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Ab initio study of He-He interactions in homogeneous electron gas
Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Ab initio study of He-He interactions in homogeneous electron gas Jinlong Wang, Liang-Liang Niu, Ying Zhang ⇑ Department of Physics, Beihang University, Beijing 100191, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 20 July 2016 Accepted 1 December 2016 Available online xxxx Keywords: Helium Electron gas Ab initio Self-trapping
a b s t r a c t We have investigated the immersion energy of a single He and the He-He interactions in homogeneous electron gas using ab initio calculations. It is found that He dislikes electrons and He-He interact via the He induced Friedel oscillations of electron densities. A critical electron density at which the global binding energy extremum shifts from the first minimum to the second one is identified. We also discover that the He-He global binding energy minimum of 0.09 eV is reached at an optimal electron density of 0.04 e/Å3, corresponding to an optimal He-He separation of 1.7 Å. Further, the He atoms are found to gain a trivial amount of 2s and 2p states from the free electrons, inducing a hybridization between the He s- and p-states. The present results can qualitatively interpret the well-known He self-trapping behavior in metals. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Helium (He), the lightest noble gas element, exhibits a closedshell electronic structure with two electrons situating at the 1s orbital, and thus is not chemically reactive with other elements. The behavior of He in metals has been under extensive study for decades because of its scientific interest and technological significance [1–10]. For instance, the retention, agglomeration and blistering of He in W can severely deteriorate the mechanical and thermal properties of tungsten (W) as the most promising plasma facing material in future nuclear fusion reactors [11–14]. To date, features such as the insolubility, high mobility and self-trapping behavior of He in metals have been well recognized. Generally, adopting the simplified ‘‘jellium model”, the dissolution of light impurities in metals (especially free-electron-like ones such as Li, Na and Mg) can be treated as them being embedded in an electron sea [15]. This model, in which the ion cores of the metals are transformed into an infinite positive background, has been applied onto various issues regarding light impurities, such as H, He and Li in metals [16–21], in metal vacancies [16,22], on metal surfaces [23–25] and in complicated nonuniform electronic systems [26]. Using similar concept, the effective medium theory [26–34], the Finnis-Sinclair method [35] and the embedded atom method [36–38], assuming that the potential energy of an atom in a metal is proportional to the electron density provided by the surrounding atoms, have proved to be quite successful in describing many-atom interactions in metals. ⇑ Corresponding author. E-mail address: [email protected] (Y. Zhang).
Despite the tremendous insights provided by these prior studies and the progress made, no results have been reported regarding He-He interactions in homogeneous gas and a fundamental understanding with respect to the origin of He self-trapping in metals is still lacking. In the present work, we have performed ab initio calculations to investigate a single He and He-He interactions in homogeneous electron gas covering a wide range of electron densities and He-He separations. The results are compared with those from He-He interactions in metals and the implications of this study are discussed. 2. Methodology Our calculations are performed within the framework of density functional theory (DFT) using the Vienna Ab initio Simulation Package (VASP) [39,40] with projector-augmented wave (PAW) potentials [41,42]. The kinetic energy cutoff for the plane-wave basis set is set to be 400 eV. The Gaussian smearing method is employed for the Fermi surface smearing with a width of 0.1 eV. The electron exchange-correlation energy is described by the local-density approximation (LDA) for He in the uniform electron gas. The convergence criteria for the electronic self-consistent iteration and the ionic relaxation are 105 eV and 102 eV/Å, respectively. Brillouin zones are sampled with a 9 9 9 k-point mesh by the Monkhorst-Pack scheme [43]. Due to the three-dimensional periodic boundary conditions used, a periodic array of simulation cells, each with a dimension of 15 15 15 Å3 containing varying densities of electrons and one or two He atoms is constructed. Considering the Friedel oscillation [44,45] of electron density induced by He, the size of the supercell is tested to ensure the
http://dx.doi.org/10.1016/j.nimb.2016.12.006 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
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J. Wang et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
uniformity of electron density at the cell borders. The immersion energy of He atoms and the diatomic He-He binding energy in the homogeneous electron gas are defined as follows.
Eim ðnHeÞ ¼ Eðqe þ nHeÞ nEðHeÞ Eðqe Þ; ðn ¼ 1 or 2Þ
ð1Þ
Eb ¼ Eim ð2HeÞ 2Eim ðHeÞ ¼ Eðqe þ 2HeÞ þ Eðqe Þ 2Eðqe þ HeÞ;
ð2Þ
wherein E(qe + He) and E(qe + 2He) are the total energies of the system with one and two He atoms immersed in the homogeneous electron gas with a density of qe, respectively, while E(He) and E (qe) are the energies of a free He atom and homogeneous electron gas with a density of qe. A negative Eb denotes attraction between two He atoms. 3. Results and discussion Fig. 1 illustrates dependence of immersion energy of a single He in an electron gas on electron density. A linear fitting of the curve gives a slope of 37.79 eV/(e/Å3), in good agreement with 40.94 eV/ (e/Å3) from Ref. [46]. The correlation can be better reflected by the binomial fitting and the presence of a small curvature (see the inset of Fig. 1) has also been observed in Ref. [46]. The key point is that He prefers to stay in regions where the electron density is lowest. The fact that He dislikes electrons can be demonstrated by the large formation energies of He in metals [2] and its strong trapping at defects with large free volume such as vacancy, vacancyimpurity complex, dislocation, grain boundary and SIA clusters in metals [4,47–51]. Utilizing free electron densities typically found in Na (0.026 e/Å3), Li (0.049 e/Å3), Mg (0.086 e/Å3) and a wide range of other densities, we plot in Fig. 2(a) the electron radial variation induced by He as a function of distance. Evidently, an effect of Friedel oscillations can be observed, the amplitude of which increases with elevating densities. Moreover, the locations of the electron density minima are found to shift towards He with increasing electron densities. Selecting the same electron densities used in Fig. 2 (a), the He-He binding energy as a function of their separation is presented in Fig. 2(b). In comparison with the case of vacuum, in which the He-He binding energy reaches the minimum (less than 0.01 eV) at an optimal distance of 2.47 Å (agreeing well with Refs. [50–52]), the He-He binding energy in the electron gas displays similar damped Friedel oscillations as Fig. 2(a). Additionally, the
Fig. 1. Immersion energy of a single He in an electron gas as a function of electron density. Linear and binomial fittings are used to describe the data.
Fig. 2. (a) Charge radial variation induced by one He and (b) He-He binding energy (Eb) as a function of separation in the electron gas with a wide range of densities. The electron densities q are in units of e/Å3.
He-He separations at the corresponding minima are also found to decrease with increasing electron densities. This effect has also been found in a previous DFT study regarding H-H interaction in an electron gas [17], though in a different manner. A closer examination of Fig. 2(a) and (b) reveals their remarkable similarity. In order to further illustrate Fig. 2 and probe how He-He interaction strength is influenced by the Friedel oscillations of electron densities, we show in Fig. 3(a) and (b) the global minimum electron density induced by one He and the He-He binding energy as a function of electron density. The corresponding distance and He-He separations are also given. We should remark that the minimum density and the binding energy here are the global minimum presented in Fig. 2(a) and (b), respectively. It is found that the He-He binding energy experiences a shift from the first minimum (separation 1.61 Å) to the second one (separation 3.59 Å) at a critical electron density of 0.08 e/Å3. Further, the binding energy extremum of 0.09 eV is reached at an electron density of 0.04 e/Å3 with an optimal He-He separation of 1.7 Å. The optimal electron density is also illustrated by the presence of the minimum in Fig. 3(c) in which He-He are fixed at a separation of 1.7 Å. In short, Figs. 2 and 3 suggest that He-He interact indirectly via the oscillated electron density. More specifically, He atoms embedded in the electron gas cause Friedel oscillations, resulting in each of them to situate at the electron density minima (according to Fig. 1) of their counterparts. By fixing He-He at separations of 1.7 and 9 Å, Fig. 4(a) presents the charge axial distribution in vacuum and in an electron gas with a density of 0.049 e/Å3. Overall, the electron gas exerts little
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
J. Wang et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
Fig. 3. (a) Variation of global minimum density induced by one He and the corresponding distance from the He with increasing electron density. (b) Variation of He-He binding energy and the corresponding separation with increasing electron density. (c) Variation of He-He binding energy with increasing electron density at a constant separation of 1.7 Å. Note that only electron densities up to 0.2 e/Å3 are shown.
influence on the charge axial distribution, though a trace of electron accumulation at He sites is observed, as illustrated by inset A. Due to the screening effect of the electron gas on the He atoms, the electron density around the He displays a trivial variation (see inset B). This is in direct contrast with the fact that H can easily pair with electrons in the electron gas [17], the present work confirms that closed-shell He is not active in interacting with electrons. Fig. 4(b) gives the projected density of states (PDOS) of He-He at fixed separations of 1.7 and 9 Å. Tiny 2s and 2p states near the Fermi energy can be observed in the inset. A degenerate 1s state locates at 16 eV can be spotted for the case of 9 Å, indicating no interaction exists between them. For the case of 1.7 Å, it is found that the 1s states become r and r⁄ molecular orbitals. The 2s, 2p states arising from the free electrons hybridize with 1s electrons, and hybridize with each other forming r2s and r2p orbitals, while two small p-states peaks at 15 and 18 eV reveal the hybridization between s and p states. The s- and p-projected DOS of He near the Fermi level have also been found in bcc transition metals [2,53]. The axial distribution of charge density difference of He-He with a fixed separation of 1.7 Å in vacuum and in an electron gas with a density of 0.049 e/Å3 is presented in Fig. 5(a) and (b), respectively. In vacuum, the electron density decreases between two He atoms and increases in the outside, implying the repulsion
3
Fig. 4. (a) Charge density distribution along the symmetry axis of He-He at fixed separations of 1.7 Å and 9 Å in vacuum and in an electron gas with a density of 0.049 e/Å3. (b) PDOS of two He with fixed separations of 1.7 Å and 9.0 Å in an electron gas with a density of 0.049 e/Å3.
Fig. 5. Axial distribution of charge density difference for He-He at a fixed separation of 1.7 Å in (a) vacuum and (b) an electron gas with a density of 0.049 e/Å3.
between them; while it is the other way around for He in the electron gas, indicating the attraction between them. Further, as shown by the inset of Fig. 5(b), the polarization of He atoms away from spherical symmetry suggests a hybridization between 1s and 2p orbitals. Specifically, a reduction of the He 1s density and an
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
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increase of 2p density (mainly 2pz) can be observed. Similar results were uncovered regarding He scattering at Rh (110) surface [54]. A wealth of studies regarding the dissolution of He in metals, including Fe, Rh, V, Nb, Ta, Mo and W, have shown that He can acquire a certain amount of s and p states near the Fermi energy level because of the hybridization between transition metals d states and He p states [2,53–55]. We should point out that, in comparison with He-He interaction (0.01 eV) in the vacuum, the He-He binding energy is increased at all electron densities (see Fig. 2(b)). For metals such as Na and Li, whose free electron densities located within the region of the ‘first minimum’ in Fig. 2(b), He can easily self-trap due to the large binding energy and small He-He equilibrium separations in this region; while in terms of metals such as Fe and W, whose free electron densities located beyond this region, it is likely that the local electron densities might be dramatically lowered due to the lattice distortion induced by He atoms, which subsequently shifts the He-He interaction region from the ‘second minimum’ to the ‘first minimum’. This qualitatively explains the He self-trapping behaviors in metals. Nevertheless, the maximum He-He binding energies are less than 0.1 eV (see Fig. 3(b)), much smaller than those found in metals. For example, the binding energies are 0.42 eV and 1.03 eV with equilibrium separations of 1.57 Å and 1.50 Å in Fe [51] and W [4], respectively. This suggests that the assumption of a homogeneous free electron density in these metals might be deficient or that the He induced electron density variation in these metals are quite different from that in a homogenous free electron gas. 4. Conclusions We have elucidated the He-He interaction features in homogenous electron gas using ab initio calculations based on DFT method. Energetically, He dislikes electrons as previously reported. It is discovered that He-He interact via the He induced Friedel oscillations of electron densities and multiple He-He binding energy minima can be identified. There is a critical electron density at which the binding energy global extremum shifts from the first minimum to the second one. Moreover, we find that He-He binding energy extremum of 0.1 eV is reached at an optimal electron density of 0.04 e/Å3, corresponding to an optimal He-He separation of 1.7 Å. Further, the He atoms are found to gain a trivial amount of 2s and 2p states from the free electrons, inducing a hybridization between He s- and p-states. The present results can qualitatively interpret the well-known He self-trapping behavior in metals. Nevertheless, the simplified jellium model fails to explain the strong He-He attraction in metals such as Fe and W on a quantitative basis, suggesting that the assumption of a homogeneous free electron density in these metals might be deficient or that He induced variation of electron density in metals might be very different from that in a homogeneous electron gas. Studies are in progress to further explore the origin of He self-trapping in metals incorporating both electronic (variation of electron density) and mechanical (lattice distortions) contributions. Acknowledgement This work is supported by the National Natural Science Foundation of China (NSFC) with No. 51671009. Ying Zhang
acknowledges the support from the Beijing Key Discipline Foundation of Condensed Matter Physics. References [1] W.D. Wilson, C.L. Bisson, M.I. Baskes, Phys. Rev. B 24 (1981) 5616–5624. [2] T. Seletskaia, Y. Osetsky, R.E. Stoller, G.M. Stocks, Phys. Rev. B 78 (2008) 134103. [3] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Appl. Phys. Lett. 87 (2005) 163113. [4] C.S. Becquart, C. Domain, Phys. Rev. Lett. 97 (2006) 196402. [5] L. Yang, X.T. Zu, H.Y. Xiao, F. Gao, H.L. Heinisch, R.J. Kurtz, K.Z. Liu, Appl. Phys. Lett. 88 (2006) 091915. [6] F. Sefta, K.D. Hammond, N. Juslin, B.D. Wirth, Nucl. Fusion 53 (2013) 073015. [7] A. Lasa, S.K. Tähtinen, K. Nordlund, EPL (Europhysics Letters) 105 (2014) 25002. [8] L. Sandoval, D. Perez, B.P. Uberuaga, A.F. Voter, Phys. Rev. Lett. 114 (2015) 105502. [9] H.-B. Zhou, Y.-L. Liu, S. Jin, Y. Zhang, G.-N. Luo, G.-H. Lu, Nucl. Fusion 50 (2010) 115010. [10] J. Wang, L.-L. Niu, X. Shu, Y. Zhang, Nucl. Fusion 55 (2015) 092003. [11] H. Ullmaier, Nucl. Fusion 24 (1984) 1039. [12] H. Trinkaus, B.N. Singh, J. Nucl. Mater. 323 (2003) 229–242. [13] S. Kajita, S. Takamura, N. Ohno, D. Nishijima, H. Iwakiri, N. Yoshida, Nucl. Fusion 47 (2007) 1358. [14] R.D. Smirnov, S.I. Krasheninnikov, Nucl. Fusion 53 (2013) 082002. [15] J.K. Nørskov, Phys. Rev. B 20 (1979) 446–454. [16] M. Manninen, P. Hautojärvi, R. Nieminen, Solid State Commun. 23 (1977) 795– 798. [17] S.A. Bonev, N.W. Ashcroft, Phys. Rev. B 64 (2001) 224112. [18] C.O. Almbladh, U. von Barth, Z.D. Popovic, M.J. Stott, Phys. Rev. B 14 (1976) 2250–2254. [19] Z.D. Popovic, M.J. Stott, Phys. Rev. Lett. 33 (1974) 1164–1167. [20] A.I. Duff, J.F. Annett, Phys. Rev. B 76 (2007) 115113. [21] V.U. Nazarov, C.S. Kim, Y. Takada, Phys. Rev. B 72 (2005) 233205. [22] J.K. Nørskov, Solid State Commun. 24 (1977) 691–693. [23] N.D. Lang, J.K. Nørskov, Phys. Rev. B 27 (1983) 4612–4616. [24] E. Zaremba, W. Kohn, Phys. Rev. B 15 (1977) 1769–1781. [25] N.D. Lang, A.R. Williams, Phys. Rev. B 18 (1978) 616–636. [26] M.J. Stott, E. Zaremba, Phys. Rev. B 22 (1980) 1564–1583. [27] J.K. Nørskov, N.D. Lang, Phys. Rev. B 21 (1980) 2131–2136. [28] K.W. Jacobsen, J.K. Norskov, M.J. Puska, Phys. Rev. B 35 (1987) 7423–7442. [29] J.K. Nørskov, K.W. Jacobsen, P. Stoltze, L.B. Hansen, Surf. Sci. 283 (1993) 277– 282. [30] M.J. Puska, R.M. Nieminen, Phys. Rev. B 43 (1991) 12221–12233. [31] K.W. Jacobsen, P. Stoltze, J.K. Nørskov, Surf. Sci. 366 (1996) 394–402. [32] M. Manninen, J.K. Norskov, C. Umrigar, J. Phys. F: Met.Phys. 12 (1982) L7. [33] J. Nørskov, Phys. Rev. Lett. 48 (1982) 1620. [34] J.K. Nørskov, Phys. Rev. B 26 (1982) 2875–2885. [35] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45–55. [36] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50 (1983) 1285–1288. [37] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983–7991. [38] M.S. Daw, Phys. Rev. B 39 (1989) 7441–7452. [39] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558–561. [40] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [41] P. Blöchl, Phys. Rev. B 50 (1994) 17953–17979. [42] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [43] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188–5192. [44] J.S. Langer, S.H. Vosko, J. Phys. Chem. Solids 12 (1960) 196–205. [45] P. Jena, K.S. Singwi, R.M. Nieminen, Phys. Rev. B 17 (1978) 301–307. [46] M.J. Puska, R.M. Nieminen, M. Manninen, Phys. Rev. B 24 (1981) 3037–3047. [47] C.S. Becquart, C. Domain, J. Nucl. Mater. 385 (2009) 223–227. [48] H.-B. Zhou, Y.-L. Liu, Y. Zhang, S. Jin, G.-H. Lu, Nucl. Instrum. Methods Phys. Res. B 267 (2009) 3189–3192. [49] G. Lucas, R. Schäublin, J. Phys.: Condens. Matter 20 (2008) 415206. [50] L. Zhang, X. Shu, S. Jin, Y. Zhang, G.H. Lu, J. Phys.: Condens. Matter 22 (2010) 375401. [51] L. Zhang, Y. Zhang, G.H. Lu, J. Phys.: Condens. Matter 25 (2013) 095001. [52] C.-C. Fu, F. Willaime, Phys. Rev. B 72 (2005) 064117. [53] T. Seletskaia, Y. Osetsky, R.E. Stoller, G.M. Stocks, Phys. Rev. Lett. 94 (2005) 046403. [54] M. Petersen, S. Wilke, P. Ruggerone, B. Kohler, M. Scheffler, Phys. Rev. Lett. 76 (1996) 995–998. [55] X.T. Zu, L. Yang, F. Gao, S.M. Peng, H.L. Heinisch, X.G. Long, R.J. Kurtz, Phys. Rev. B 80 (2009) 054104.
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Ab initio study of He-He interactions in homogeneous electron gas Jinlong Wang, Liang-Liang Niu, Ying Zhang ⇑ Department of Physics, Beihang University, Beijing 100191, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 20 July 2016 Accepted 1 December 2016 Available online xxxx Keywords: Helium Electron gas Ab initio Self-trapping
a b s t r a c t We have investigated the immersion energy of a single He and the He-He interactions in homogeneous electron gas using ab initio calculations. It is found that He dislikes electrons and He-He interact via the He induced Friedel oscillations of electron densities. A critical electron density at which the global binding energy extremum shifts from the first minimum to the second one is identified. We also discover that the He-He global binding energy minimum of 0.09 eV is reached at an optimal electron density of 0.04 e/Å3, corresponding to an optimal He-He separation of 1.7 Å. Further, the He atoms are found to gain a trivial amount of 2s and 2p states from the free electrons, inducing a hybridization between the He s- and p-states. The present results can qualitatively interpret the well-known He self-trapping behavior in metals. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Helium (He), the lightest noble gas element, exhibits a closedshell electronic structure with two electrons situating at the 1s orbital, and thus is not chemically reactive with other elements. The behavior of He in metals has been under extensive study for decades because of its scientific interest and technological significance [1–10]. For instance, the retention, agglomeration and blistering of He in W can severely deteriorate the mechanical and thermal properties of tungsten (W) as the most promising plasma facing material in future nuclear fusion reactors [11–14]. To date, features such as the insolubility, high mobility and self-trapping behavior of He in metals have been well recognized. Generally, adopting the simplified ‘‘jellium model”, the dissolution of light impurities in metals (especially free-electron-like ones such as Li, Na and Mg) can be treated as them being embedded in an electron sea [15]. This model, in which the ion cores of the metals are transformed into an infinite positive background, has been applied onto various issues regarding light impurities, such as H, He and Li in metals [16–21], in metal vacancies [16,22], on metal surfaces [23–25] and in complicated nonuniform electronic systems [26]. Using similar concept, the effective medium theory [26–34], the Finnis-Sinclair method [35] and the embedded atom method [36–38], assuming that the potential energy of an atom in a metal is proportional to the electron density provided by the surrounding atoms, have proved to be quite successful in describing many-atom interactions in metals. ⇑ Corresponding author. E-mail address: [email protected] (Y. Zhang).
Despite the tremendous insights provided by these prior studies and the progress made, no results have been reported regarding He-He interactions in homogeneous gas and a fundamental understanding with respect to the origin of He self-trapping in metals is still lacking. In the present work, we have performed ab initio calculations to investigate a single He and He-He interactions in homogeneous electron gas covering a wide range of electron densities and He-He separations. The results are compared with those from He-He interactions in metals and the implications of this study are discussed. 2. Methodology Our calculations are performed within the framework of density functional theory (DFT) using the Vienna Ab initio Simulation Package (VASP) [39,40] with projector-augmented wave (PAW) potentials [41,42]. The kinetic energy cutoff for the plane-wave basis set is set to be 400 eV. The Gaussian smearing method is employed for the Fermi surface smearing with a width of 0.1 eV. The electron exchange-correlation energy is described by the local-density approximation (LDA) for He in the uniform electron gas. The convergence criteria for the electronic self-consistent iteration and the ionic relaxation are 105 eV and 102 eV/Å, respectively. Brillouin zones are sampled with a 9 9 9 k-point mesh by the Monkhorst-Pack scheme [43]. Due to the three-dimensional periodic boundary conditions used, a periodic array of simulation cells, each with a dimension of 15 15 15 Å3 containing varying densities of electrons and one or two He atoms is constructed. Considering the Friedel oscillation [44,45] of electron density induced by He, the size of the supercell is tested to ensure the
http://dx.doi.org/10.1016/j.nimb.2016.12.006 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
2
J. Wang et al. / Nuclear Instruments and Methods in Physics Research B xxx (2016) xxx–xxx
uniformity of electron density at the cell borders. The immersion energy of He atoms and the diatomic He-He binding energy in the homogeneous electron gas are defined as follows.
Eim ðnHeÞ ¼ Eðqe þ nHeÞ nEðHeÞ Eðqe Þ; ðn ¼ 1 or 2Þ
ð1Þ
Eb ¼ Eim ð2HeÞ 2Eim ðHeÞ ¼ Eðqe þ 2HeÞ þ Eðqe Þ 2Eðqe þ HeÞ;
ð2Þ
wherein E(qe + He) and E(qe + 2He) are the total energies of the system with one and two He atoms immersed in the homogeneous electron gas with a density of qe, respectively, while E(He) and E (qe) are the energies of a free He atom and homogeneous electron gas with a density of qe. A negative Eb denotes attraction between two He atoms. 3. Results and discussion Fig. 1 illustrates dependence of immersion energy of a single He in an electron gas on electron density. A linear fitting of the curve gives a slope of 37.79 eV/(e/Å3), in good agreement with 40.94 eV/ (e/Å3) from Ref. [46]. The correlation can be better reflected by the binomial fitting and the presence of a small curvature (see the inset of Fig. 1) has also been observed in Ref. [46]. The key point is that He prefers to stay in regions where the electron density is lowest. The fact that He dislikes electrons can be demonstrated by the large formation energies of He in metals [2] and its strong trapping at defects with large free volume such as vacancy, vacancyimpurity complex, dislocation, grain boundary and SIA clusters in metals [4,47–51]. Utilizing free electron densities typically found in Na (0.026 e/Å3), Li (0.049 e/Å3), Mg (0.086 e/Å3) and a wide range of other densities, we plot in Fig. 2(a) the electron radial variation induced by He as a function of distance. Evidently, an effect of Friedel oscillations can be observed, the amplitude of which increases with elevating densities. Moreover, the locations of the electron density minima are found to shift towards He with increasing electron densities. Selecting the same electron densities used in Fig. 2 (a), the He-He binding energy as a function of their separation is presented in Fig. 2(b). In comparison with the case of vacuum, in which the He-He binding energy reaches the minimum (less than 0.01 eV) at an optimal distance of 2.47 Å (agreeing well with Refs. [50–52]), the He-He binding energy in the electron gas displays similar damped Friedel oscillations as Fig. 2(a). Additionally, the
Fig. 1. Immersion energy of a single He in an electron gas as a function of electron density. Linear and binomial fittings are used to describe the data.
Fig. 2. (a) Charge radial variation induced by one He and (b) He-He binding energy (Eb) as a function of separation in the electron gas with a wide range of densities. The electron densities q are in units of e/Å3.
He-He separations at the corresponding minima are also found to decrease with increasing electron densities. This effect has also been found in a previous DFT study regarding H-H interaction in an electron gas [17], though in a different manner. A closer examination of Fig. 2(a) and (b) reveals their remarkable similarity. In order to further illustrate Fig. 2 and probe how He-He interaction strength is influenced by the Friedel oscillations of electron densities, we show in Fig. 3(a) and (b) the global minimum electron density induced by one He and the He-He binding energy as a function of electron density. The corresponding distance and He-He separations are also given. We should remark that the minimum density and the binding energy here are the global minimum presented in Fig. 2(a) and (b), respectively. It is found that the He-He binding energy experiences a shift from the first minimum (separation 1.61 Å) to the second one (separation 3.59 Å) at a critical electron density of 0.08 e/Å3. Further, the binding energy extremum of 0.09 eV is reached at an electron density of 0.04 e/Å3 with an optimal He-He separation of 1.7 Å. The optimal electron density is also illustrated by the presence of the minimum in Fig. 3(c) in which He-He are fixed at a separation of 1.7 Å. In short, Figs. 2 and 3 suggest that He-He interact indirectly via the oscillated electron density. More specifically, He atoms embedded in the electron gas cause Friedel oscillations, resulting in each of them to situate at the electron density minima (according to Fig. 1) of their counterparts. By fixing He-He at separations of 1.7 and 9 Å, Fig. 4(a) presents the charge axial distribution in vacuum and in an electron gas with a density of 0.049 e/Å3. Overall, the electron gas exerts little
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
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Fig. 3. (a) Variation of global minimum density induced by one He and the corresponding distance from the He with increasing electron density. (b) Variation of He-He binding energy and the corresponding separation with increasing electron density. (c) Variation of He-He binding energy with increasing electron density at a constant separation of 1.7 Å. Note that only electron densities up to 0.2 e/Å3 are shown.
influence on the charge axial distribution, though a trace of electron accumulation at He sites is observed, as illustrated by inset A. Due to the screening effect of the electron gas on the He atoms, the electron density around the He displays a trivial variation (see inset B). This is in direct contrast with the fact that H can easily pair with electrons in the electron gas [17], the present work confirms that closed-shell He is not active in interacting with electrons. Fig. 4(b) gives the projected density of states (PDOS) of He-He at fixed separations of 1.7 and 9 Å. Tiny 2s and 2p states near the Fermi energy can be observed in the inset. A degenerate 1s state locates at 16 eV can be spotted for the case of 9 Å, indicating no interaction exists between them. For the case of 1.7 Å, it is found that the 1s states become r and r⁄ molecular orbitals. The 2s, 2p states arising from the free electrons hybridize with 1s electrons, and hybridize with each other forming r2s and r2p orbitals, while two small p-states peaks at 15 and 18 eV reveal the hybridization between s and p states. The s- and p-projected DOS of He near the Fermi level have also been found in bcc transition metals [2,53]. The axial distribution of charge density difference of He-He with a fixed separation of 1.7 Å in vacuum and in an electron gas with a density of 0.049 e/Å3 is presented in Fig. 5(a) and (b), respectively. In vacuum, the electron density decreases between two He atoms and increases in the outside, implying the repulsion
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Fig. 4. (a) Charge density distribution along the symmetry axis of He-He at fixed separations of 1.7 Å and 9 Å in vacuum and in an electron gas with a density of 0.049 e/Å3. (b) PDOS of two He with fixed separations of 1.7 Å and 9.0 Å in an electron gas with a density of 0.049 e/Å3.
Fig. 5. Axial distribution of charge density difference for He-He at a fixed separation of 1.7 Å in (a) vacuum and (b) an electron gas with a density of 0.049 e/Å3.
between them; while it is the other way around for He in the electron gas, indicating the attraction between them. Further, as shown by the inset of Fig. 5(b), the polarization of He atoms away from spherical symmetry suggests a hybridization between 1s and 2p orbitals. Specifically, a reduction of the He 1s density and an
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006
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increase of 2p density (mainly 2pz) can be observed. Similar results were uncovered regarding He scattering at Rh (110) surface [54]. A wealth of studies regarding the dissolution of He in metals, including Fe, Rh, V, Nb, Ta, Mo and W, have shown that He can acquire a certain amount of s and p states near the Fermi energy level because of the hybridization between transition metals d states and He p states [2,53–55]. We should point out that, in comparison with He-He interaction (0.01 eV) in the vacuum, the He-He binding energy is increased at all electron densities (see Fig. 2(b)). For metals such as Na and Li, whose free electron densities located within the region of the ‘first minimum’ in Fig. 2(b), He can easily self-trap due to the large binding energy and small He-He equilibrium separations in this region; while in terms of metals such as Fe and W, whose free electron densities located beyond this region, it is likely that the local electron densities might be dramatically lowered due to the lattice distortion induced by He atoms, which subsequently shifts the He-He interaction region from the ‘second minimum’ to the ‘first minimum’. This qualitatively explains the He self-trapping behaviors in metals. Nevertheless, the maximum He-He binding energies are less than 0.1 eV (see Fig. 3(b)), much smaller than those found in metals. For example, the binding energies are 0.42 eV and 1.03 eV with equilibrium separations of 1.57 Å and 1.50 Å in Fe [51] and W [4], respectively. This suggests that the assumption of a homogeneous free electron density in these metals might be deficient or that the He induced electron density variation in these metals are quite different from that in a homogenous free electron gas. 4. Conclusions We have elucidated the He-He interaction features in homogenous electron gas using ab initio calculations based on DFT method. Energetically, He dislikes electrons as previously reported. It is discovered that He-He interact via the He induced Friedel oscillations of electron densities and multiple He-He binding energy minima can be identified. There is a critical electron density at which the binding energy global extremum shifts from the first minimum to the second one. Moreover, we find that He-He binding energy extremum of 0.1 eV is reached at an optimal electron density of 0.04 e/Å3, corresponding to an optimal He-He separation of 1.7 Å. Further, the He atoms are found to gain a trivial amount of 2s and 2p states from the free electrons, inducing a hybridization between He s- and p-states. The present results can qualitatively interpret the well-known He self-trapping behavior in metals. Nevertheless, the simplified jellium model fails to explain the strong He-He attraction in metals such as Fe and W on a quantitative basis, suggesting that the assumption of a homogeneous free electron density in these metals might be deficient or that He induced variation of electron density in metals might be very different from that in a homogeneous electron gas. Studies are in progress to further explore the origin of He self-trapping in metals incorporating both electronic (variation of electron density) and mechanical (lattice distortions) contributions. Acknowledgement This work is supported by the National Natural Science Foundation of China (NSFC) with No. 51671009. Ying Zhang
acknowledges the support from the Beijing Key Discipline Foundation of Condensed Matter Physics. References [1] W.D. Wilson, C.L. Bisson, M.I. Baskes, Phys. Rev. B 24 (1981) 5616–5624. [2] T. Seletskaia, Y. Osetsky, R.E. Stoller, G.M. Stocks, Phys. Rev. B 78 (2008) 134103. [3] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Appl. Phys. Lett. 87 (2005) 163113. [4] C.S. Becquart, C. Domain, Phys. Rev. Lett. 97 (2006) 196402. [5] L. Yang, X.T. Zu, H.Y. Xiao, F. Gao, H.L. Heinisch, R.J. Kurtz, K.Z. Liu, Appl. Phys. Lett. 88 (2006) 091915. [6] F. Sefta, K.D. Hammond, N. Juslin, B.D. Wirth, Nucl. Fusion 53 (2013) 073015. [7] A. Lasa, S.K. Tähtinen, K. Nordlund, EPL (Europhysics Letters) 105 (2014) 25002. [8] L. Sandoval, D. Perez, B.P. Uberuaga, A.F. Voter, Phys. Rev. Lett. 114 (2015) 105502. [9] H.-B. Zhou, Y.-L. Liu, S. Jin, Y. Zhang, G.-N. Luo, G.-H. Lu, Nucl. Fusion 50 (2010) 115010. [10] J. Wang, L.-L. Niu, X. Shu, Y. Zhang, Nucl. Fusion 55 (2015) 092003. [11] H. Ullmaier, Nucl. Fusion 24 (1984) 1039. [12] H. Trinkaus, B.N. Singh, J. Nucl. Mater. 323 (2003) 229–242. [13] S. Kajita, S. Takamura, N. Ohno, D. Nishijima, H. Iwakiri, N. Yoshida, Nucl. Fusion 47 (2007) 1358. [14] R.D. Smirnov, S.I. Krasheninnikov, Nucl. Fusion 53 (2013) 082002. [15] J.K. Nørskov, Phys. Rev. B 20 (1979) 446–454. [16] M. Manninen, P. Hautojärvi, R. Nieminen, Solid State Commun. 23 (1977) 795– 798. [17] S.A. Bonev, N.W. Ashcroft, Phys. Rev. B 64 (2001) 224112. [18] C.O. Almbladh, U. von Barth, Z.D. Popovic, M.J. Stott, Phys. Rev. B 14 (1976) 2250–2254. [19] Z.D. Popovic, M.J. Stott, Phys. Rev. Lett. 33 (1974) 1164–1167. [20] A.I. Duff, J.F. Annett, Phys. Rev. B 76 (2007) 115113. [21] V.U. Nazarov, C.S. Kim, Y. Takada, Phys. Rev. B 72 (2005) 233205. [22] J.K. Nørskov, Solid State Commun. 24 (1977) 691–693. [23] N.D. Lang, J.K. Nørskov, Phys. Rev. B 27 (1983) 4612–4616. [24] E. Zaremba, W. Kohn, Phys. Rev. B 15 (1977) 1769–1781. [25] N.D. Lang, A.R. Williams, Phys. Rev. B 18 (1978) 616–636. [26] M.J. Stott, E. Zaremba, Phys. Rev. B 22 (1980) 1564–1583. [27] J.K. Nørskov, N.D. Lang, Phys. Rev. B 21 (1980) 2131–2136. [28] K.W. Jacobsen, J.K. Norskov, M.J. Puska, Phys. Rev. B 35 (1987) 7423–7442. [29] J.K. Nørskov, K.W. Jacobsen, P. Stoltze, L.B. Hansen, Surf. Sci. 283 (1993) 277– 282. [30] M.J. Puska, R.M. Nieminen, Phys. Rev. B 43 (1991) 12221–12233. [31] K.W. Jacobsen, P. Stoltze, J.K. Nørskov, Surf. Sci. 366 (1996) 394–402. [32] M. Manninen, J.K. Norskov, C. Umrigar, J. Phys. F: Met.Phys. 12 (1982) L7. [33] J. Nørskov, Phys. Rev. Lett. 48 (1982) 1620. [34] J.K. Nørskov, Phys. Rev. B 26 (1982) 2875–2885. [35] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45–55. [36] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50 (1983) 1285–1288. [37] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983–7991. [38] M.S. Daw, Phys. Rev. B 39 (1989) 7441–7452. [39] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558–561. [40] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [41] P. Blöchl, Phys. Rev. B 50 (1994) 17953–17979. [42] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [43] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188–5192. [44] J.S. Langer, S.H. Vosko, J. Phys. Chem. Solids 12 (1960) 196–205. [45] P. Jena, K.S. Singwi, R.M. Nieminen, Phys. Rev. B 17 (1978) 301–307. [46] M.J. Puska, R.M. Nieminen, M. Manninen, Phys. Rev. B 24 (1981) 3037–3047. [47] C.S. Becquart, C. Domain, J. Nucl. Mater. 385 (2009) 223–227. [48] H.-B. Zhou, Y.-L. Liu, Y. Zhang, S. Jin, G.-H. Lu, Nucl. Instrum. Methods Phys. Res. B 267 (2009) 3189–3192. [49] G. Lucas, R. Schäublin, J. Phys.: Condens. Matter 20 (2008) 415206. [50] L. Zhang, X. Shu, S. Jin, Y. Zhang, G.H. Lu, J. Phys.: Condens. Matter 22 (2010) 375401. [51] L. Zhang, Y. Zhang, G.H. Lu, J. Phys.: Condens. Matter 25 (2013) 095001. [52] C.-C. Fu, F. Willaime, Phys. Rev. B 72 (2005) 064117. [53] T. Seletskaia, Y. Osetsky, R.E. Stoller, G.M. Stocks, Phys. Rev. Lett. 94 (2005) 046403. [54] M. Petersen, S. Wilke, P. Ruggerone, B. Kohler, M. Scheffler, Phys. Rev. Lett. 76 (1996) 995–998. [55] X.T. Zu, L. Yang, F. Gao, S.M. Peng, H.L. Heinisch, X.G. Long, R.J. Kurtz, Phys. Rev. B 80 (2009) 054104.
Please cite this article in press as: J. Wang et al., Ab initio study of He-He interactions in homogeneous electron gas, Nucl. Instr. Meth. B (2016), http://dx. doi.org/10.1016/j.nimb.2016.12.006