Influence of Krypton Seeding on EU DEMO Operation with Lithium Divertor

DEMO reactor with liquid lithium divertor operation in presence of krypton seeding is analyzed. Integrated core-scrape-off layer-divertor code COREDIV is used. Sputtering model is revised and sputtering dependence on the divertor surface temperature fitted to the experimental data is added. It was found that Kr seeding is beneficial for DEMO operation. Plasma fuel dilution by lithium ions is reduced to the level of 1%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\%$$\end{document} for high level of Kr seeding and fusion power is comparable to the modelling results for argon or neon seeded DEMO with tungsten divertor. Simulations show that high confinement mode is possible in presence of Kr seeding (power across the separatrix stays above the L-H threshold).


Introduction
Modelling of DEMO reactor with liquid lithium divertor indicates [1] that the goal of a divertor power load reduction may be achieved by the fuel ions dilution in the core and consequently the reduction of the fusion power. Effectively, DEMO operation with liquid lithium divertor is not excluded but results in lower fusion gain factor, Q.20 compared to Q ' 35 in the case of tungsten divertor with additional neon or argon seeding [2]. In DEMO reactor with a tungsten divertor a significant part of the fusion power in the core is dissipated by means of radiation, what allows for reducing divertor heat load. The predicted fusion gain value in not seeded DEMO with tungsten divertor may be close to 40. However, estimated value of the target power load in such a scenario is above the technological limit for a tungsten divertor. Seeding of medium Z impurities may improve power dissipation in the SOL but for the price of a lower fusion gain value. Contrary to tungsten divertor seeding of medium and high Z impurities in DEMO with lithium divertor might be beneficial from the fusion gain point of view. They are able to dissipate power both in the core and in the SOL (see Fig. 1 for radiative capabilities of Li, Kr and W), reducing divertor power load and, in effect, decreasing lithium dilution. This feature however comes at a cost of higher risk of radiative collapse and loosing operation in a high confinement mode (H-mode).
The aim of this work is to analyze operation of a DEMO reactor with liquid lithium divertor in which cooling of the plasma is achieved by impurity seeding where Kr (Z=36) is chosen as seeded gas. Sputtering model is revised and data verified thermal sputtering component is added in order to cover a range of divertor surface temperature from the melting point up to the level where evaporation is a main source of lithium. Conditions in the reactor core and scrape-off layer and their dependence on the Kr concentration are analyzed by means of the code COREDIV [3,4]. The paper is organized as follows, modelling approach is described in Sect. 2, sputtering and evaporation models are described in Sect. 3

Modelling Approach
Modelling of fusion reactor plasma from the point of view of energy balance requires coupling between the core and the scrape-off layer (SOL) plasma. Intrinsic impurity sources like sputtering and evaporation depend on the particle and heat fluxes hitting the divertor plate. On the other hand presence of impurities in the core affects the fusion performance either by dilution (low Z impurities) or by additional radiation (medium and high Z impurities). The coupling between the core and the SOL requires the transport problem to be addressed in both regions simultaneously.

COREDIV
COREDIV is an integrated core-SOL-divertor code. The code self-consistently solves 1D radial transport equations of plasma and impurities in the core region and 2D multifluid transport based on Braginskii-like equations in the scrape-off layer region. Since this work is a continuation of work described in Ref. [1] we restrict here to listing the parameters used in the simulation and to the description of the sputtering model updated for the purpose of this study and refer the reader to Ref. [1] and references therein for more detailed description of the code. The value of radial anomalous transport in the SOL D SOL rad ¼ 0:42 m 2 =s. Thermal conductivities in the SOL are set to v e;i ¼ 0:18 m 2 =s. Values for D SOL rad and v e;i are chosen to be the same as in TECXY simulations for DEMO with liquid metal divertor [5]. Sputtering, prompt redeposition and evaporation of lithium is calculated self-consistently, depending on the plasma conditions in the divertor region. Since the evaporation model finds the liquid metal surface temperature it was straightforward to add to the physical sputtering module a component dependent on the target surface temperature.

Lithium Source
Sputtering Physical sputtering model implemented in COREDIV is based on revised Bohdansky semi-empirical formula [6,7] describing collisional sputtering on solid surfaces at normal incidence where s KrC n ðeÞ is the Kr-C nuclear stopping cross section [8] and e is the reduced energy M 1 (Z 1 ) and M 2 (Z 2 ) are atomic weights (atomic numbers) of the projectile and the target respectively and a L is the Lindhard screening length given by and square of the electron charge e 2 ¼ 14:4 eVÅ . The values of the parameter Q and the thershold energy E th for D?, He?, and Li? incident species are extracted from Ref. [9]. For other incident species like krypton the formula for the Q parameter and for the threshold energy is based on the formulas by Garcia-Rosales et al. [10]. A separate formula is used for the sputtering yield dependence on the angle of incidence a 0 [11] Yða 0 Þ Yð0Þ where f is a fitted parameter and a opt is the angle at which the sputtering yield reaches maximum. Values of parameters f and a opt were calculted using formulas found in Ref. [11] with the exception of D?, He? and Li? sputtering where the value of f was extracted from Ref. [9]. Experiment by Allain at al. [9] in which the D?, He? and Li? sputtering was measured were performed on D saturated Li target. In order to reproduce experimental results one has to include a model of preferential sputtering. Otherwise the sputtering yield calculated from formulas (1) and (4) provide overestimated sputtering yield values. Similarly as in Ref. [9] an expression derived by Andersen and Bay [12] was used where c D and c Li are concentrations of D and Li atom on the lithium matrix and U D and U Li are the surface escape energy of each of the component. For simplicity it is Sputtering on solid targets depends on the incident particle energy. However, experiments performed on liquid lithium targets showed strong sputtering dependence on the liquid metal surface temperature [13,14]. In the considered DEMO divertor conditions the heat flux to the divertor target may heat the surface far above the melting point of the target but still below the point where evaporation is dominant. Therefore, it was necessary to include the thermal sputtering component along with the collisional one. In order to implement in COREDIV the sputtering dependence on the target surface temperature a function Y fit th ðTÞ was added to the collisional sputtering yield where E 0 ¼ 300 eV; 500 eV and 350 eV for Dþ, Heþ and Liþ sputtering respectively. Formula from Eq. (4) is used for the angular dependence f ða 0 Þ. The thermal sputtering component dependence on the divertor surface temperature is based on the adatom model [15] Y fit th ðTÞ ¼ where the fitted parameters Y ad ; A and E eff are listed in Table 1.
The fitted curves plotted against the experimental data are shown in Fig. 3. The solid black line corresponds to the self-sputtering and the solid red and blue lines correspond to helium and deuterium sputtering respectively.
Since the measurement of thermal sputtering yield was done at the incindent angle a 0 ¼ 45°C, throughout the simulation the same value of a 0 was used. However, in order to facilitate the possiblity of angle dependent calculations in the future the available data was extrapolated to the values of the incident angle other than 45°C using the following formula The dependence on the angle of incidence is factored out and the Li particle flux leaving the divertor plate can be written as where C inc is the flux of incident particles, Y normal phys ¼ Y col ðE 0 Þ þ Y th ðTÞ and C vap is the evaporation rate. Uisng this set of parameters the self-sputtering yield is less than unity with maximum value of Y ¼ 0:97 at T ¼ 750C°C and E 0 ¼ 165 eV.  Evaporation A one dimensional LM divertor scheme is presented in Fig. 4. The divertor consists of a metal mesh or 3-D printed structure wetted with liquid metal and a solid basis which is in contact with coolant. It is assumed that the heat flow across the divertor is one dimensional. For the purpose of this study, the metal mesh material and solid basis material is assumed to be tungsten. Temperatures T x and T s are the substrate-mesh contact surface temperature and the liquid metal surface temperature respectively. A liquid metal surface is assumed to be static. Heat flux q is an input and evaporation rate C is an output of the model. Parameters d mesh and d W are a liquid metal mesh layer and a substrate layer thicknesses respectively. Lithium and tungsten thermal conductivities k i ðTÞ, i ¼ Li, W for reference values at 800 K are given in Table 2. Tungsten and lithium thermal conductivities dependence on temperature was extracted from Ref. [16]. Liquid metal and mesh thermal conductivity is an average of tungsten and lithium thermal conductivities. Coolant temperature T 0 is a parameter of the model and is kept constant during the simulations. Heat flux q is split into the heat flux q cool which is transferred without losses further to the bottom of the divertor plate and into latent heat of evaporation P vap , Latent heat of evaporation is proportional to the evaporation rate C, P vap ¼ h vap C vap , where the coefficient h vap is a latent heat of evaporation per one particle. Temperatures T x and T s are the substrate-mesh contact surface temperature and the liquid metal surface temperature respectively. A flux of particles C ½m À2 s À1 leaving the divertor surface as a function of surface temperature is given by the Hertz-Knudsen equation.
where saturated vapor pressure p sv ¼ 5:055 À 8023=T s [17]. The dependence of the evaporation rate on the liquid metal surface tempearature is close to the values presented in Refs. [18] and [19]. Surface temperature depends on the heat flux q cool reaching the divertor bottom via the Fourier equation q cool ¼ ÀkðTÞrT. Eventually, the evaporation rate is calculated by solving Eq. (10), C vap ¼ ðq À q cool Þ=h vap .
In    Fig. 5 Lithium evaporation rate dependence on the heat flux (solid) and the evaporation rate in the case when latent heat of evaporation is set to zero (dashed) asymptotic limit C ¼ q=h vap and at the boiling point it becomes purely linear. This feature is responsible for saturation of the evaporation rate dependence on the heat flux allowing for higher fluxes and stabilizes the numerical behavior of the code.
Throughout the simulatins the value of the density at the separatrix was set to n sep ¼ 3:27 Â 10 19 m À3 . The source of puffed impurity is located at 0.9 fraction of the distance from the stagnation point to divertor plate. It is a gaussian source with the half width of 0.05 (again as a fraction of the distance from the stagnation point to divertor plate). The X-point is located at 0.83 of this distance, hence the puffing is applied already in the divertor region but closer to the Xpoint than to the divertor plate. Krypton as an inert gas is a highly recycling impurity and a position of source has minor influence on the SOL plasma. As for the evaporation model, the tungsten substrate depth was set to d W ¼ 10 mm, liquid lithium mesh thickness d Li ¼ 1 mm and the temperature of the divertor bottom T 0 ¼ 200 C.

Kr Seeding
Kr seeding was applied starting from the gas puff rate C puff Kr ¼ 10 16 s À1 and it was increased up till the electron temperature at the plate stayed above 1eV or up till the code converged. Increasing krypton radiation in the core and in the SOL may eventually outbalance the heating energy which results in a radiational collapse in real conditions. In such a situation the corediv solver does not find the solution for the energy balance equation.
Dependence of impurities (Li,He and Kr) concentrations in the core on the krypton gas puff rate is given in Fig. 6 (top). As expected Li concentration steadily decreases from 13:1% down to 1% of the total electron concentration. Kr concentration increases from practically zero to 0:134%. Also He concentration increases as it's a product of increasing number of DT reactions. It is interesting to notice that the fuel dilution (Fig. 6, middle) in the center of the core defined as a ratio ðn 0 e À n 0 i Þ=n 0 e (n 0 e;i being electron and ion concentration in the center of the core) increases first with the Kr gas puff rate reaching maximum at C Kr puff ' 10 20 s À1 . This is due to the fact that for small values of Kr gas puff ion density does not decrease as fast as electron density. The observed drop in electron density is mainly due to the fact that electron temperature in the center (T 0 e , bottom of the Fig. 6) increases in result of the energy released in the increasing number of fusion reactions. Fusion gain is defined as Q ¼ 5 P a =P aux , where P a is the energy carried by a particles and P aux is the external heating power. Slight increase in fuel dilution for low values (C Kr puff \10 20 s À1 ) of the gas puff can explain why Q increases first slowly and from C Kr puff ' Â10 20 s À1 its slope becomes steeper and Q reaches 33.7 for the maximum gas puff level. The effect is more pronounce when P a is plotted against the Kr concentration (C Kr ) in the core (see Fig. 7). Power balance in the core is shown in Fig. 7.Dashed green line (P LH ) marks the minimum P SOL value necessary for the operation in the high confinement regime. At low level of Kr concentration radiation in the core increases faster than a power. This causes decrease of P SOL to the value of the L-H transition. The value of P SOL reaches minimum of 114 MW at C Kr ¼ 0:055%. At this point the slope of the heating power becomes steeper that the slope of core radiation curve and at higher level of Kr concentration (C Kr [ 0:08%) power across the separatrix is higher than L-H threshold.Changes in the power balance with krypton concentration are reflected in the increase of the radiation In the case of no Kr seeding lithium radiates ( see Fig. 9, dashed) predominantly directly in front of the divertor plate. When Kr seeding is applied the spike is replaced by a broader and smaller profile of Kr radiation (red) located further from the divertor plate. Li radiation (black, solid) is becomes much weaker and its maximum is aligned with the Kr radiation one.
As was observed in [1] evaporation was the dominant source of lithium in the plasma. However, when thermal sputtering is included, self-sputtering and evaporation contribute equally at low level of Kr concentration (see Fig. 10). Deuterium sputtering is order of magnitude lower than self-sputtering. Level of the Li influx rate (C Li ' 2 Â 10 24 s À1 ) and the maximum surface temperature (T max surf [ 750 C) found in the simulations (at low Kr seeding values) are comparable to those where thermal   sputtering was neglected (see Ref. [1]). The noticable difference is that the plasma conditions in the divertor region at low Kr seeding found in the simulations when thermal sputtering is included correspond to the conditions found in Ref. [1] for higher values of tungsten substrate depth (d W ' 15 À 20 mm) rather than d W ¼ 10 mm used in this paper. One may conclude that including thermal sputtering results in high evaporation regime taking place for lower values of the LMD tungsten substrate depth.
In the case of pure lithium DEMO reactor predicted value of the divertor surface tempearture (above 700 C) is higher than the numbers reported for example by [18] and [19]. This can be mostly attributed to the size of the lithium source, which is of the order of 10 24 s À1 and is possible due to the volume of DEMO reactor and due the level of fusion power which is not dissipated by means of other impurities (which is the subject of this paper). Note, that in presence of krypton seeding the total Li source is lower but not significantly and stays at the level of 5 Â 10 23 s À1 despite the fact that the surface temperature drops down to values below 400 C. As it was already noted, in the case of high Kr seeding the dominant source of lithium is sputtering.
Although power to the plate drops only by 30% with krypton concentration, conditions in the divertor region changes significantly with Kr seeding. Maximum power flux to the plate (P max plate ) drops from above 7 MW=m 2 down to the value of 2 MW=m 2 ( see Fig. 10). Maximum divertor surface temperature (T max surf ) also drops from 755 C down to 350 C which results in a decline of the evaporation rate by few orders of magnitude. The total lithium influx to the plasma decreases with Kr concentration by 80% down to values C Li $ 3:6 Â 10 23 s À1 for the maximum C Kr value. The hierarchy of the lithium sources changes. Deuterium sputtering contribution increases and equals self-sputtering at C Kr ¼ 0:075%. Self-sputtering drops down to the level of $ 2 Â 10 22 s À1 which is 6% fraction of D sputtering) Electron temperature at the plate (T plate e ) drops from 14 eV down to the level of 2eV and ion concentration at the plate increases significantly marking reduction of the lithium source (see Fig. 11).Maximum Kr concentration achieved during the simulations was C Kr ¼ 0:145% at C puff Kr ¼ 1:46 Â 10 21 s À1 . Any attempts to further increase the gas puffing ended up in a collapse of the run. This may indicate a partial thermal collapse of the edge plasma as observed in [19] and therefore set the upper limit on Kr impurity concentration. This is however a very rough estimate taking into account limitations of the model used in this work to model extreme conditions of the scrape-off layer plasma. Factors which may influence operation at temperatures close to divertor detachment such as shape of the divertor, pumping efficency and private region are not taken into account.

Conclusions
Lithium source model implemented in COREDIV was revised and thermal sputtering component ws added in order to stay in agreement with data available for sputtering on deuterium saturated lithium surface. Applying Kr seeding to a EU DEMO scenario with liquid lithium divertor helps in improving the power performance of the reactor. Fusion gain Q increases from values below 20 up the level above 30. Lithium dilution is significantly attenuated. Li concentration in the reactor core plasma drops from values above 13% down to around 1%. Increasing fusion power balances krypton radiation in the core and power across the separatrix stays above the L-H threshold. However, in the mid values of Kr concentration P SOL stays only slightly above the L-H threshold. Kr seeding is also beneficial for the divertor power load. Maximum power load is kept at low level (\8 MW=m 2 ) being reduced to 2 MW=m 2 for the maximum krypton concentration level. Reduction in the divertor power load is accompanied by the decrease of the divertor surface temperature from the level of 700 C to the level below 400 C. Low temperature at the divertor plate and high level of SOL radiation for high values of Kr concentration may indicate that reactor operation at very high level of Kr seeding might be difficult to control.
Acknowledgements This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.