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Nuclear Instruments and Methods in Physics Research B 212 (2003) 96-100 www.elsevier.com/locate/nimb

Theoretical description of fast proton scattering from steel surface under grazing incidence
N.V. Novikov
a

a,*

, Ya.A. Teplova

a,*

, Yu.A. Fainberg

a,*

, V.A. Kurnaev

b

Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia b Moscow Engineering and Physics Institute, Moscow 115409, Russia

Abstract The angle-energy distributions of fast protons scattered from a steel surface under grazing incidence are studied theoretically. The presented model is based on the Monte-Carlo method and on calculations of the total collision crosssections using the Ziegler stopping tables. The energy distributions at a fixed scattered angle are characterized by certain parameters. The 300 and 500 keV proton calculation results for a 4À grazing angle are in qualitative agreement with the authorså experimental data. ã 2003 Elsevier B.V. All rights reserved.
PACS: 34.40.Dy Keywords: Angle-energy distributions; Grazing incidence; Fast protons

1. Introduction The experiments [1,2] with fast ion reflection from metal surfaces have shown that the scattered ion energy E0 is distributed within a broad range of 0 6 E0 6 E0 , where E0 P 100 keV is the incident ion energy. In the case of slow ion-surface collisions, the energy distribution is deprived of the narrow maximum at E0 % E0 associated with the processes in a thin layer of a medium [3]. The binary collision theory attributes this narrow maximum to the processes in a thin layer, wherein only a few ion- atom collisions occur. An incident ion is known to lose energy ranging from a few eV to dozens of eV

Corresponding authors. E-mail address: teplova@anna19.npi.msu.su (N.V. Novikov).

*

in a single ion-atom collision. This fact, together with the large width of the energy distribution (0 6 E0 6 E0 ) in the fast ion-surface collisions, suggests two alternative theoretical approaches to explaining the resultant distribution form. The first approach assumes that an incident ion interacts with a few atoms of the medium simultaneously. In the grazing incidence geometry, a scattered fast ion cannot break through the repulsive collective potential of surface and fails to penetrate into a solid [4]. When moving parallel with a surface, an ion gets scattered by the surface plasmons [5]. The ion stopping power theory in the impact parameter representation [6], as well as the dielectric function formalism [7], can qualitatively explain the interaction of an incident ion with a free-electron gas. The second approach assumes that an ion penetrates into the solid and that its collisions with

0168-583X/$ - see front matter ã 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01484-8


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atoms of the medium occur repeatedly. The computer simulation of a projectile transport is usually made by the Monte-Carlo technique, which is also used to describe ejection of atoms of medium through an ion-bombarded surface [8] and to reproduce the charge distribution of the ions that traversed a thin foil [9]. The simulation codes are based on different models of projectile interaction with a screened target atom potential. This work is aimed at Monte-Carlo analyzing the energy-angle distributions of fast protons reflected from a steel surface and at comparing theory and experiment [1,2].

cade initiated by ion bombardment [10,11] and in the collisions of ions with foil [12]. 2. The target atom species is calculated from the distribution of probabilities qm =q. 3. The reaction channel for projectile collision with a target atom m is calculated from the disP tribution of probabilities rmj ?Eî= j rmj ?Eî. After that, the energy emj Ì E Ð E0 transferred to the atom is determined. 4. The scattering angle h ?0 6 h 6 pî is calculated from the differential cross-section drmj ?Eî=dX Ì rmj ?Eîfmj ?E; hî=?2pî; Zp fmj ?E; hî sin?hî dh Ì 1;
0

?4î

2. Theory Let an incident particle of energy E0 move in an amorphous medium of density, X X qÌ qm Ì q0m NA =lm ; ? 1î
m m

where fmj ?E; hî is the angular distribution function. 5. The azimuth angle u ?0 6 u 6 2pî is calculated in terms of the equiprobable approximation. Now that the direction and energy of a scattered particle are known, the point of the next collision can be determined. The history of a single incident particle finishes when the energy E0 goes below Emin , or the particle escapes from the medium. If the number of the incident particle histories is great, we can find the energy-angle distributions of the scattered projectiles. The input parameters in the simulation are the densities qm , the cross-sections rmj ?Eî, and the angular functions fmj ?E; hî. It should be noted that the crosssections rmj ?Eî may depend on qm . In this work we treat the reflection of fast protons from a steel surface at a small angle a. The steel is considered to consist approximately of 70% iron and some quantities of nickel and chromium. The difference between the proton stopping powers of iron and nickel, or of iron and chromium, is below 3% at E < 30 keV and below 0.5% at E > 30 keV [13]. Therefore, we can very well use the pure iron approximation. This means that the subscript m has the same value in (1)-(4). The total crossP section j rmj ?E î is calculated from the experimental stopping power [13]. The fractions of the P elastic, rm;elas ?Eî= j rmj ?Eî, and inelastic, P rm;inelas ?Eî= j rmj ?Eî, channels coincide with the same fractions in proton-helium collisions [14,15].

where the subscript m characterizes the atom specie of medium; qm , q0m are the densities of the atoms in units of 1/cm3 and g/cm3 , respectively; lm is the mass of target atom in g/mol; NA is the Avogadro number 1/mol. The Monte-Carlo method offers the following steps to simulate the projectile transport in a medium between two collisions. 1. The ion motion direction is known. Therefore, the point of each next collision can be calculated from the track length L. The probability of the first collision of a projectile between L and L ? dL along its line of flight is expressed as P ?Lî dL Ì exp?Ðqrtot ?EîLîqrtot ?Eî dL; X X rtot ?Eî Ì qm =q rmj ?Eî;
m j

? 2î ? 3î

where rtot ?Eî is the total cross-section of ionmedium interaction, rmj ?Eî is the reaction cross-section in the jth channel (ionization, excitation, elastic scattering, charge transfer, etc.) for collision of an ion with the mth atom. The Monte-Carlo simulation by distribution (2) was used in a sputtering from a collision cas-


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The transmitted energy emj Ì E Ð E0 of inelastic collision equals the ionization potential of a target atom. The present theoretical model disregards the charge transfer channels, but allows approximately for the H? +Fe ! H0 + Fe? ! H? +Fe process by including the proton absorption crosssection at E < 20 keV. The angular functions fmj ?hî of elastic and inelastic scattering reactions are taken to be the same and can be presented as fmj ?E; hî Ì C ?Eî=?h2 ? v?Eî2 î; ? 5î

3. Calculation results We have calculated the energy-angle distributions of the 300 and 500 keV protons reflected from the steel surface. The incident proton grazing angle is a Ì 4À. Up to 4 Ç 106 histories of incident particles were used at the two incidence energies. At a fixed scattering angle h, the energy distributions were analyzed using the parameters Emax ?hî, k ?hî, W ?hî and a?hî. The values of the parameters were derived from the theoretical and experimental [1,2] distributions of the reflected protons at h > a. The parameter Emax ?hî decreases with increasing h, Fig. 1. This is quite explainable. The probability for a projectile ion to be reflected from a surface at large angles h is proportional to the projectile track length in a semi-infinite medium, or to the number of collisions. The projectile ion energy decreases in each collision. Therefore, the energy, at which the reflected ion distribution is peaking, decreases with increasing h. The relative amplitude parameter (6) is plotted in Fig. 2. Both experiment and theory show that this parameter is peaking near h Ì 2a. The calculation results indicate that, at large angles h, the parameter k ?hî is higher for 500 keV protons than for 300 keV protons, in agreement with the experimental data. At small angles h, the 300-500 keV

where C ?Eî is the normalization factor in (4); the parameter v?Eî is the angular distribution width at energy E and decreases with increasing E. The v?Eî value is determined by variation method. The Monte-Carlo algorithms, the generator of pseudorandom numbers and the estimated relative errors are all from [16]. We characterize the angle-energy distribution F ?E; hî by certain parameters. The first, Emax ?hî, is the parameter of the reflected proton energy at which the distribution F ?E; h Ì constî=E0 is peaking. The number of reflected ions R dE0 F ?E0 ; hî is peaking at h Ì 2a. Therefore, this particular scattering direction was selected to normalize the energy distribution amplitude. The parameter k ? hî Ì F ? E
max

? hî ; hî =F ? E

max

? 2aî ; 2aî

? 6î

is the relative amplitude of energy distribution. Note that k ?2aî Ì 1. The parameter W ?hî Ì ?E? ?hîÐ EÐ ?hîî=E
0

? 7î

is the full width at half maximum (FWHM) ?0 < W ?hî < 1î, where the energies E? ?hî and EÐ ?hî in (7) are determined as F ?E
max

? h î ; h î = 2 Ì F ? E? ? h î ; h î ;

E? ? h î > E

max

? hî ; ? 8î

F ?E

max

? h î ; h î = 2 Ì F ? EÐ ? h î ; h î ;

EÐ ? h î < E

max

? hî : ? 9î
Fig. 1. The parameter Emax ?hî that characterizes the energy at which the reflected proton energy distribution in collisions with the steel surface at a small angle a Ì 4À is peaking. The present theoretical results: the solid curve is the calculations for the 500 keV incident protons; the broken curve is the calculations for the 300 keV incident protons. The dots N and d are the experimental data [1,2] at 500 and 300 keV, respectively.

The asymmetry parameter a? hî Ì ? E
max

?hîÐ EÐ ?hîî=?E? ?hîÐ E

max

?hîî

is the ratio of the left-hand and right-hand half widths. This parameter vanishes in the limit of large scattering angles.


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Fig. 2. The amplitude parameter k ?hî of the reflected proton energy distribution in collisions with steel surface at angle a Ì 4À. The notation is the same as in Fig. 1.

calculation result ratio is reversed, contrary to the experimental data. Understanding the disagreement necessitates further research. There is the qualitative agreement between the theoretical and the experimental results for the FWHM parameter (7), Fig. 3. Note that the W ?hî variation is small within the range of scattering angles where W ?hî=2 < Emax ?hî. In the case of W ?hî=2 > Emax ?hî, W ?hî decreases rapidly at large h. Fig. 4 presents the calculation results and the experimental data for the asymmetry parameter (9). The decrease in a?hî with increasing h is found at any h both theoretically and experimentally.

Fig. 4. The asymmetry parameter a?hî of the reflected proton energy distribution in collisions with steel surface at angle a Ì 4À. The notation is the same as in Fig. 1.

4. Conclusion We studied the angular and energy distributions of fast proton reflected from steel surface at grazing incidence angles. The presented Monte-Carlo simulation is based on the experimental stopping power [13]. The calculation results obtained are in good agreement with experimental data.

References
[1] Ya.A. Teplova, Yu.A. Fainberg, V.S. Kulikauskas, Surface N3 (1997) 72 (in Russian). [2] Yu.A. Fainberg, Ya.A. Teplova, V.S. Kulikauskas, Surface N4 (2002) 43 (in Russian). [3] V.A. Kurnaev, E.S. Mashkova, V.A. Molchanov, The Reflection of Light Ions by Solid Surface, Energoizdat, Moscow, 1985. [4] H. Winter, J. Remillieux, J.C. Poizat, Nucl. Instr. and Meth. B 48 (1990) 382. [5] A.A. Lucas, Phys. Rev. B 20 (1979) 4990. [6] P. Sigmund, Nucl. Instr. and Meth. B 125 (1997) 77. [7] N.R. Arista, Phys. Rev. A 49 (1994) 1885. [8] P. Sigmund, M.T. Robinson, M. Hautala, et al., Nucl. Instr. and Meth. B 36 (1989) 110 (and the references therein). [9] H. Ogawa, N. Sakamoto, N. Shiomi-Tsuda, H. Tsuchida, Phys. Rev. A 61 (2000) 032717. [10] A. Gras-Marti, H.M. Urbassek, N.R. Arista, F. Flores, Interaction of Charge Particles with Solid and Surfaces, Plenum Press, NY, 1991, 744p.

Fig. 3. The FWHM parameter W ?hî of the reflected proton energy distribution in collisions with steel surface at angle a Ì 4À. The notation is the same as in Fig. 1.


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