The calculation method of interaction between metal atoms under influence of the radiation

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1 IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS The calculation ethod of interaction between etal atos under influence of the radiation To cite this article: S N Yanin 015 IOP Conf. Ser.: Mater. Sci. Eng Related content - Fast Calculation Method for the Rate Equations of Plasa Utaro Furukane and Toshiatsu Oda - Managing the Zabezi River Qi Gao and Shaoxia Yang - Research on the shear of reinforced concrete T beas based on the MCFT calculation ethod Feng Haoxiong and Yi Weijian View the article online for updates and enhanceents. This content was downloaded fro IP address on 5/0/019 at :01

2 The calculation ethod of interaction between etal atos under influence of the radiation 1. Introduction S N Yanin 1 1 Tosk, 64050, Russia, Tosk Polytechnic University E-ail: yanin@tpu.ru Abstract. A ethod of calculation of interatoic interaction potentials in the presence of ionized states has been developed. They have been obtained for the atos with different ionization degree on exaple of aluinu. The Heine-Abarenkov-Anialu odel potential for factors was eployed. The for factor paraeters of ionized atos was deterined on the base of the quantu defect ethod using the atoic-spectroscopy data. The potential of interatoic interaction for different charged states with different degree of ionization were deterined. It is known that intensive radiation generates ionized atos within the track of a fast charged particle as well as near the solid surface. The potentials of interactions between these atos and surrounding particles change considerably. The knowledge of corresponding potentials of interatoic interaction is necessary for studying the behavior of partially or entirely ionized atter. Therefore, aluinu was taken as an exaple studying the interatoic potentials in etal containing ionized states. Calculations were based on the ethod of pseudopotentials using Heine-Abarenkov- Anialu odel potentials with paraeters which were deterined fro spectroscopic ters of free ions following the ethod of quantu defect.. Quantu defect ethod The energetic levels E nl of a single electron in the area of positive ions with the electron shells being siilar to the spherically syetric electron shells of inert gases, in particular Na +, Mg +, Al +, Si 4+, P 5+,S 6+, Cl 7+, are known to satisfy the relation [1]: / n, (1) nl nl nl where Z is the ion-core charge, δ nl and Δ nl denotes quantu defects characterizing the Ze charge induced deviation of an atoic potential in such a configuration of inert gas shells fro a Coulob ones. Figure1 shows a linear dependence of the value of spectroscopic ters of the atoic series considered on a square of the ion-core charge. It turns out that there also are siilar dependences for additionally ionized cores (figure). Here the following series Na +, Mg +, Al 4+, Si 5+, P 6+,S 7+, Cl 8+ will be realized. Using this epirical law we have deterined the ode potential paraeters of the additionally ionized cores with respect to the energies ε nl =E nl Δ nl, as it was done for transition etals in [1]. Radial wave equation with a odel potential describing the electron otion is written as Content fro this work ay be used under the ters of the Creative Coons Attribution.0 licence. Any further distribution of this work ust aintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 1 d l( l 1) 1 v ( r ) ( r ), l dr r () Figure 1. Spectroscopic ter values Ep as function of the cheical valence. where χ(r) is the radial wave function and l is the angular oentu. With such a representation ε is given in rydbergs and vl in atoic units. The odel potential of the electron ion-core interaction has a for V A ( ) for r R l l V Z / r for r R, l () here R is a odel radius closed to the ion-core radius which provides for joining the inside and outside solutions of the wave equation. After aking a substitution of Zr and r A into expression () it will be written as l l l for r R, (4) l( l 1) 0 where / z. for r R, (5) Figure. Spectroscopic ter values E p as function of the ion-core charge. Solution of equation (4) are the spherical Bessel functions and equation (5) is the Coulob wave one which was solved according to the procedure described in []. By setting the logarithic derivatives fro solutions of equation (4) equal to those in equation (5) at r=r one obtains an expression for deterining the paraeters A l :

4 0 l 1 l l 1 xj ( x) D ( ZR ) ( N, l ) D ( ZR ) l, (6 j U ( ZR ) ( N, l) U ( ZR ) 0 l 1 l l where x R А, j l(x) denotes the spherical Bessel functions, 0 U l, 1 U l are regular and irregular l nl parts of the Coulob wave function respectively, N=n- nl expresses an effective quantu nuber, D l = (du l /d), l 1 Г ( N l ) N tg ( N -l -1) ( N, l), (7) Г ( N l 1) where Г is a gaa-function []. The values of spectroscopic ters = nl were calculated fro tables [4]. Al values as function of energy = nl at fixed l. Al ay be calculated precisely only for the energies which are consistent with the lines observed in the electron eission spectra of a core potential. There in the function A l( nl) is chosen so that the odel potential will give eaten values consistent with nl. Therefore, first, we estiate A l for as any energy values as experiental lines exist, after wards linear interpolation add extrapolation to the Feri energies is ade. Here the question arises, what value is a real energy of the conduction electrons. In contrast to isolated ion (equation) Schrodinger equation for the conduction electron in etal has a for ( V ) ( V ) ε, ion rest (8) where (V+) ion is the potential of this ion and (V+) rest is the potential caused by other ions and conduction electrons. Expression (8) ay be written as ( V ) ( V ) ε, ion rest (9) Equation (9) is siilar to equation () for an isolated ion with electron energy E ef ( V S ), (10) F rest where k F 0,6Ze I I E E, F i с x с * x с 5 (11) R a Figure. Energetic dependence of the paraeters Al for l=0 and z=. Dashes region corresponds to the energies of the occupied conduction electron states. I i is the ionization energy of outside electron in a free ato, I c is the electron cohesive energy, k F is the Feri radius, * is the effective electron ass, Е х is the exchange energy, Е с is the correlation energy of a free electronic gas per an electron, х and с are respectively, exchange and correlation potentials, and R a is an atoic radius.

5 The potential (V+) rest induced by the rest residual ions and conduction electrons ay be written as [5]: rest = х + с, (1) Ze R V. rest (1) R 4 R a a Figure shows an energetic dependence of paraeter A for an electron in the field of a threecharged ion. The energy levels nl in Al -ion, the Feri energy F 0 and that of the botto of conduction band k / are shown in ters of the ion energy [6]. 0 F F The ion-core radius R is an iportant value for calculating interatoic potential. Its procedure c estiation is following. According to [7,8] the wave electron function outside the core has a for n l 1 / ) L n 1 ( n l 1)! exp( n l!, where L is the added Lagerr polynoial, *=Zr/na 0, a 0 is a Bohr radius. Then the wave function will be, (14) n k l * k ( 1) n 1! / ).. (15) k 0 ( n k)! (k )! k! ( n )! ni n n l! exp( Inside the core it is Arj ( ). (16) ni i Here A is a noralization factor, yr, y A i ni. Expression (15) included the nubers up to (n-) order, * n and * n are the ajor ones. If keep only these two ebers, one ay think that n adopts any fractional values. At logarithic derivatives fro expressions (15) and (16) ay be set equal : 1 1 n y. * * ( n 1)( n ) na 1 ctg 0 r R c the By solving graphically equation (17) we hand found the ion-core radius of additionally ionized ato. As it should be guess R c value seeed to be not very sensitive to the degree of ionization. * As for an effective ass and charge of electron conduction, we counted / 1, as it was done * in [1], while the paraeter inserted in expression to define the effective charge e e1, which results fro orthogonality correction, was calculated fro a forulae / /. (17) R c R a (18) Table 1. Paraeters of odel potential Z A0 A1 A R Rc 1,8 1,64 1,9 1,08 0,04 4 1,5 1,756 1,46 1,05 0,0 4

6 Table 1 illustrates the paraeters of a odel potential for usually charged aluinu, taken fro reference [1], and those obtained by us for additionally ionized core for coparison. All the easured values are in atoic units.. Results and discussion Figure 4 represents the potential of interatoic interaction for different charged states in aluinu which were estiated using the above data. Calculation was done for a situation when the ionized ato concentration is sall and consequently the conduction electron concentration does not differ fro an usual one. Figure 4. Interatoic potential: 1 Al + and Al + ; Al 4+ and Al + ; Al 4+ and Al 4+. This figure shows that ionization leads to the strong decrease in the depth of the first iniu of the potential function corresponding to the distance area between the nearest neighbors for three- or four-charged ions. Moreover, for the pair of particles with four charges the first iniu disappears absolutely. Atos fall to the repulsion branch of interaction potential. As a result, the crystal lattice changes to the state of nonequilibriu. On the basis of the siilar potentials of interatoic interactions, applying the ethod of olecular dynaics, we odeled the dispersion of aluinu surface within the track of charged particles. It was shown that ionization greatly affects the dispersion process of surfaces of etals. However, the olecular-dynaic dispersion odeling in any cases is qualitative. For the ore thorough investigation of destruction processes of surfaces we should solve the equation of theral elasticity, where in its turn we should know the equation of substance condition. References [1] Anialy A O E 197 Phys. Rev. B 8 54 [] Kertis A R 1969 The wave function of Coulob (Moscow: Coputer Center of USSA Acadey of Sciences) p 14 [] Ha F S 1955 Solid State Physics vol 1 (New York) p 17 [4] Moore C E 1949 Atoic Energy Levels vol 1 (Washington National Bureau of Standarts) p 10 [5] AnialyA O E and Heine V Phys. Mag [6] Ziesche P and Lehan G 1984 Progress of the Electronic Theory of Metals vol 1 (Moscow: Mir) p 14 [7] Landau L D and Lifshits E M 1989 Theoretical Physics vol (Moscow: Nauka) p 15 [8] Harrison W 1989 Electronic Structure and Properties of Solids vol (Moscow: Mir) p 114 5