Nonperturbative Shell Correction to the Bethe Bloch Formula for the Energy Losses of Fast Charged Particles

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1 ISSN , JETP Letters, 20, Vo. 94, No., pp. 5. Peiades Pubishing, Inc., 20. Origina Russian Text V.I. Matveev, D.N. Makarov, 20, pubished in Pis ma v Zhurna Eksperimenta noi i Teoreticheskoi Fiziki, 20, Vo. 94, No., pp Nonperturbative She Correction to the Bethe Boch Formua for the Energy Losses of Fast Charged Partices V. I. Matveev and D. N. Makarov Pomor State University, Arkhangesk, Russia e-mai: matveev.victor@pomorsu.ru Received Apri 2, 20 A simpe method incuding nonperturbative she corrections has been deveoped for cacuating energy osses on compex atoms. The energy osses of fast highy charged ions on neon, argon, krypton, and xenon atoms have been cacuated and compared with experimenta data. It has been shown that the incusion of the nonperturbative she corrections noticeaby improves agreement with experimenta data as compared to cacuations by the Bethe Boch formua with the standard corrections. This undoubtedy heps to reduce the number of fitting parameters in various modifications of the Bethe Boch formua, which are usuay determined semiempiricay. DOI: 0.34/S INTRODUCTION The Bethe Boch formua underies cacuations of energy osses of fast charged partices coiding with various targets. Incuding additiona corrections, the standard Bethe Boch formua (see, e.g., []) is used in the form (the atomic system of units is used) S = 4π---N Z2. () a ( L Bethe + ΔL Boch + ΔL She + ΔL Barkas ) Here, nd v are the charge and veocity of the projectie; N a is the number of eectrons in the target; L Bethe = n(2 /I), where I is the average potentia of the ionization of the target, was cacuated by Bethe [2] in the owest order of perturbation theory; ΔL Boch = Reψ( + iz/v) + ψ(), where ψ(x) is the ogarithmic derivative of the Γ function, is the Boch correction [3]; ΔL She is the she correction (see, e.g., [4]); and ΔL Barkas is the Barkas correction [5]. A number of fitting parameters whose vaues are obtained semiempiricay are introduced in Eq. () appied to cacuate energy osses on various targets and of stopping ions moving in various media. Such a parameterization is primariy necessary for determining the effective charges of ions moving in various media. Recommended average ionization potentias of mutieectron targets are obtained by fitting the experimenta data. Corrections ΔL Barkas and ΔL She aso require a semiempirica parameterization [, 6]. In this situation, the introduction of additiona corrections whose genera form is obtained ab initio makes it possibe to hope that the number of necessary semiempirica parameters wi be reduced or their vaues wi be coser to theoretica estimates. One of the possibe additiona corrections to the Boch Bethe formua appears as foows [7]. The derivation of the Boch Bethe formua is based [3, 8, 9] on the decomposition of the entire pane of the impact parameter into two regions: the region of arge impact parameters, where perturbation theory is appicabe, and the region of sma impact parameters, where the eectrons of the target are considered as free. In this case, the contribution from the region of intermediate impact parameters (comparabe with the characteristic size of the atom) is taken into account ony in the first order of perturbation theory, athough the interaction of the projectie with the eectrons of the target is maxima in this region of the impact parameters and the nonperturbative consideration is generay necessary [6, 7]. Energy osses of fast charged partices coiding with atoms were recenty considered [7] in the eikona approximation, which makes it possibe to cacuate the effective stopping throughout the region of the impact parameters using a unified nonperturbative approach. It is shown that the nonperturbative contribution to the effective stopping from the region of impact parameters comparabe with the characteristic sizes of the eectron shes of the target can be significant as compared to the she corrections ΔL She to the Bethe Boch formua, which are cacuated in the first order of perturbation theory. For this reason, in [7, 0], it was proposed to suppement the right-hand side of Eq. () with nonperturbative correction ΔL, which can be cacuated by Eqs. (57) and (58) from [0] in the eikona approximation for a mutieectron target. Direct cacuations by these formuas are possibe ony in numerica form. Such cacuations for compex atoms as targets are very engthy and hardy possibe. Rea cacuations have been performed ony for hydrogen atoms [7, 0].

2 2 MATVEEV, MAKAROV In this work, we deveop a simpe method for cacuating energy osses on compex atoms with nonperturbative she corrections using their vaues obtained for hydrogen-ike atoms. The energy osses of fast highy charged ions on neon, argon, krypton, and xenon atoms are cacuated and compared with experimenta data. It is shown that the incusion of the nonperturbative she corrections noticeaby improve agreement with experimenta data as compared with cacuations by the Bethe Boch formua with standard corrections given by Eq. (). 2. NONPERTURBATIVE SHELL CORRECTION FOR HYDROGEN-LIKE ATOMS According to [7, 0], the nonperturbative she correction for the hydrogen atom in state 0 before a coision in the eikona approximation has the form ΔL = I 0 ( q)i* ( q) ( 2π) 3 q 0 q q ( π)2 d 2 q ΔL Boch. q 2 (2) Here, q = 2v and q 0 are the imits of integration over the momentum transfer q; therefore, q 0 = 0 can be taken (see discussion after Eq. (6) in [0]): I( q) ( iqb ) b+ s i2η = b i2η exp b b 2 d2 b, (3) where s is the projection of the coordinates of the eectron in the hydrogen atom on the impact parameter pane b. The foowing simpe approximation of Eq. (2) was obtained earier [0] (with a reative error of no more than 3%) from the representation of the scattering of Couomb waves: ΔL = γ + K 0 ( 2x) + n( x). (4) Here, γ = is the Euer constant, K 0 (2x) is the modified Besse function of the second kind, and x = (2β) /2 η/v, where η = Z/v is the Couomb parameter. The ony parameter depending on the form of function 0 in approximation (4) is parameter β, which is reated to the characteristic size b 0 of state 0 as b 0 = / β, so that parameter β is generay known ony in order of magnitude. Parameter β can be accuratey determined from the requirement on the coincidence of numerica resuts obtained by Eqs. (2) and (3) with ΔL vaues given by Eq. (4). In this manner, the vaue β = 0.4 at which Eq. (4) reproduces ΔL vaues for the hydrogen atom in the ground state was obtained in [0]. Correspondingy, b 0 = / β = According to [0], inequaity x specifies the region of param- eters nd v in which correction ΔL is sma, because the modified Besse function of the second kind in this case has the form K 0 (2x) = [γ + n(x)] with an accuracy to term x 2 nx). Using x = 0.53η/v, we concude that nonperturbative she correction ΔL shoud be taken into account under the condition η/(2v), i.e., η (because v ). This concusion is stricty vaid ony for atoms with a sma (about unity) number of eectrons, because nonperturbative she corrections for compex atoms are proportiona to the number of eectrons on the shes (see beow) and can be noticeabe even for sma x vaues. Using Eq. (2), we first cacuate ΔL for the hydrogen atom that is initiay in arbitrary state nm (where n is the principa quantum number, is the orbita anguar momentum, and m is the projection of the atter) and determine the corresponding nonperturbative she correction ΔL = ΔL nm. Then, we obtain the correction average over the orbita anguar momenta and their projections by the formua n m = ΔL n = --- ΔL nm. n 2 = 0 m = (5) Equating the cacuated ΔL n vaues to Eq. (4), we numericay obtain the β vaues for the hydrogen atoms that are initiay in states with different n vaues. These β vaues are denoted as β n ; i.e., β n are the β vaues with which Eq. (4) reproduces the ΔL n vaues. As in [0], we perform cacuations for wide ranges of the reative coision veocity and charge of the incident ion. As a resut, we obtain the foowing β n vaues, which wi be used in subsequent cacuations: β n = = 0.4, β n = 2 = , β n = 3 = , β n = 4 = , β n = 5 = (6) For the hydrogen-ike atom with the effective charge 2 of the nuceus, β n shoud be changed to β n in view of the reation b 0 = / β. Correspondingy, x = (2β n ) /2 η/v in Eq. (4). A more detaied description with parameter β coud be given by introducing β nm vaues with which Eq. (4) woud reproduce ΔL = ΔL nm vaues. However, in a aforementioned cases, such description gives resuts differing by ess than %. For this reason, we wi use beow ess detaied β n vaues given in Eqs. (6). JETP LETTERS Vo. 94 No. 20

3 NONPERTURBATIVE SHELL CORRECTION TO THE BETHE BLOCH FORMULA 3 3. NONPERTURBATIVE SHELL CORRECTION FOR COMPLEX ATOMS Using the resuts obtained for hydrogen-ike atoms, we find the nonperturbative she correction for a compex mutieectron atom. The average excitation energy of an atom coiding with an ion is written as Δε = ( ε n ε 0 )W n, n (7) where W n is the probabiity of the transition of the atom from state 0 with energy ε 0 to state n with energy ε n as a resut of coision with a stopping ion. The mutieectron atom is described with the foowing simpifying assumptions. Eectrons are considered distinguishabe. Their states are described by singeeectron wavefunctions in a mean fied. In this case, W n = W n, n2, n3,, nn, where ni are the quantum numbers of the ith eectron in singe-eectron state ni with energy ni. This state and its energy depend on the configuration of the remaining N a eectrons of the atom. According to [0], the main contribution to correction ΔL comes from high momentum transfers q (see discussion after Eq. (6) in [0]). According to [] (see the end of section 49), the atom is ionized at high momentum transfers, so that amost the entire momentum q and energy are transferred to one atomic eectron. Thus, we estimate correction ΔL taking into account ony singe-eectron excitations. In this case, average energy osses are represented in the form of the sum of osses on each individua eectron at frozen positions of the remaining atomic eectrons. Therefore, Eq. (7) can be represented in the form Δε = Δ n W n, 0,, 0 n + Δ n2 W 0, n2, 0,, 0 + +Δ nn W 0 0 nn n2,,,, (8) where W 0,, 0, ni, 0,, 0 is the probabiity of the ith eectron from initia state 0i with energy 0i to an arbitrary state ni with energy ni when the positions of the remaining atomic eectrons are frozen and Δ ni = ni 0i is the corresponding transit Each term in Eq. (8) describes energy osses on a one-eectron atom whose eectron is in the mean fied created by a remaining frozen eectrons. Average energy osses on each one-eectron atom depend on the energies and wavefunctions of a excited states ni. However, quantity ΔL of interest depends ony on the wavefunction of the initia state 0i according to Eq. (2) (see aso [0]). Thus, quantity N a ΔL for the compex atom containing N a eectrons is obtained as the sum of the ΔL nn vaues for a one-eectron atoms corresponding to terms in Eq. (8). The states of the mutieectron atom wi be described beow using hydrogen-ike functions nm ) with effective charges determined by the rues (simiar to the known Sater rues [2]) proposed in [3]. In this case, the nonperturbative she corrections ) cacuated by Eq. (4) with effective charges depend on n and and are independent of m. For this reason, they are denoted as ΔL n,. The fina formua for cacuating the nonperturbative she correction for the compex mutieectron atom has the form ΔL = ---- N n N a n,, ΔL n,. (9) Here, N n, is the number of atomic eectrons in the states with quantum numbers n and, summation is performed ony over the fied states, and sum, N n n, gives N a, i.e., the tota number of eectrons in the atom (reca that N n, are the occupation numbers for the atom that is in the ground state before coision). She corrections ΔL n, are cacuated by Eq. (4), which can be represented in the convenient form ΔL n ) (, γ K 0 2 2β n) /2 Z ) a η = v ( 2β + n ) /2 Z ) a η n v, ( ) where is the effective charge of the atomic nuceus for the eectron in state n, when the states of the remaining atomic eectrons are fixed. Note that athough β n vaues are sma according to Eq. (6), the argument of the modified Besse function of the second kind in Eq. ( cannot generay be considered sma compared to unity. For this reason, the dependence of ΔL n, on Z,, and v is compex. For competeness, we consider the behavior of ΔL n, when the argument of function K 0 (2x) is sma and the function can be represented as K 0 (2x) = n(xe γ ) x 2 n(xe γ ). Then, under the condition 2(2β n ) /2 ) Z Z/ a is ) expressed in terms of Z,, and v in the form ( ) /2 Z ) a Z β ΔL n n, = ( 2β n ) /2 Z ) a Ze γ n () JETP LETTERS Vo. 94 No. 20

4 4 MATVEEV, MAKAROV Fig.. Energy osses of krypton ions on neon atoms versus Fig. 2. Energy osses of krypton ions on argon atoms versus Fig. 3. Energy osses of krypton ions on krypton atoms versus Fig. 4. Energy osses of krypton ions on xenon atoms versus However, correction ΔL n, can make a noticeabe contribution to Eq. (9) even in this case, because it is mutipied by occupation numbers N n,. For definiteness, ΔL is cacuated for an argon atom with 8 eectrons distributed in the ground state as (s 2, 2s 2, 2p 6, 3s 2, 3p 6 ). We cacuate ΔL, 0 by Eq. ( with β n = = 0.4 from Eqs. (6) and effective (, charge = 7.65 for the active s eectron according to the rues from [3] with the fixed states of the remaining 7 eectrons of the argon atom. Then, ΔL 2, 0 is cacuated by Eq. ( with β n =2 = from ( 2, Eqs. (6) and effective charge = 4.75 for the active 2s eectron by rues from [3] with the fixed states of the remaining 7 eectrons of the argon atom. After that, ΔL 2, is cacuated by Eq. ( with β n =2 = ( 2, ) from Eqs. (6) and effective charge = 2.65 for the active 2p eectron by rues from [3] with the fixed states of the remaining 7 eectrons of the argon atom. Simiary ΔL 3, 0 and ΔL 3, are cacuated ( 3, with β n =3 = for the active 3s ( = 8.85) ( 3, ) and 3p ( = 7.05) eectrons, respectivey. As a resut, the nonperturbative she correction for the argon atom with 8 eectrons in the ground state is given by the expression ΔL = ---( 2ΔL, 0 + 2ΔL ΔL 3 0 +, 6ΔL 3, 6ΔL 2,, ). (2) 4. CALCULATION RESULTS AND COMPARISON WITH EXPERIMENTAL DATA Thus, the effective stopping of an ion with charge Z moving with veocity v on an atom containing N a eectrons has the form (see aso [0]) JETP LETTERS Vo. 94 No. 20

5 NONPERTURBATIVE SHELL CORRECTION TO THE BETHE BLOCH FORMULA 5 S = 4π Z2 ---N a ( L Bethe + ΔL Boch + ΔL She + ΔL Barkas + ΔL ), (3) where ΔL is the nonperturbative correction given by Eq. (9). In order to iustrate the necessity of incuding the nonperturbative she correction, we cacuated energy osses of fast ions on neon, argon, krypton, and xenon atoms and compared the resuts with experimenta data. To minimize the number of fitting parameters, we used the effective charge Z of stopping ion in the form [, 4, 5], which is in agreement with Bohr s estimate [6, 7], 2/3 Z = Z 0 [ exp( v/z 0 )], where Z 0 is the charge of the bare ion. Correction ΔL She was cacuated without fitting parameters by Eq. (9') from [8] based on the harmonic osciator mode with frequency ω = I, where I is the target ionization potentia. The Barkas potentia was cacuated according to [9, 20] with the recommended vaue of the singe fitting parameter κ = 2 [, 20] (notation used in [9]). The average ionization potentias of the targets were taken from [2] (Tabe VI, recommended vaues I). The resuts of cacuating energy osses by Eq. (3) incuding nonperturbative she correction ΔL are shown by the soid ines in Figs. 4, where the dotted ines are energy osses cacuated by standard formua () disregarding correction ΔL, stars are the experimenta data taken from [22], circes are the experimenta data taken from [23], and squares are the experimenta data taken from [24, 25]. The experimenta data reported in [22 25] can be aso found in It is noteworthy that the incusion of nonperturbative she correction ΔL noticeaby improves agreement with experimenta data in spite of the minimum number of fitting parameters. This work was supported by the Counci of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schoos, project no. MK REFERENCES. J. F. Zieger, App. Phys. A: Mater. Sci. Process. 85, 249 (999). 2. H. A. Bethe, Ann. Phys., Lpz. 5, 324 ( F. Boch, Ann. Phys. 6, 285 (933). 4. H. Bichse, Phys. Rev. A 65, (2002). 5. W. H. Barkas, W. Birnbaum, and F. M. Smith, Phys. Rev. 0, 778 (956). 6. P. K. Sigmund, Specia issue on Ion Beam Science: Soved and Unsoved Probems, Ed. by P. Sigmund, Dan. Vidensk. Sesk. Mat. Fys. Medd. 52, 557 (2006). 7. V. I. Matveev, D. N. Makarov, and E. S. Gusarevich, JETP Lett. 92, 28 ( J. Lindhard and A. Sorensen, Phys. Rev. A 53, 2443 (996). 9. V. J. Khodyrev, Phys. B 33, 5045 ( V. I. Matveev, D. N. Makarov, and E. S. Gusarevich, J. Exp. Theor. Phys. 2, 756 (2.. L. D. Landau and E. M. Lifshitz, Course of Theoretica Physics, Vo. 3: Quantum Mechanics: Non-Reativistic Theory (Nauka, Moscow, 989, 4th ed.; Pergamon, New York, 977, 3rd ed.). 2. J. C. Sater, Phys. Rev. 36, 57 ( G. Burns, J. Chem. Phys. 4, 52 (964). 4. L. C. Northciffe, Phys. Rev. 20, 744 ( N. J. Carron, An Introduction to the Passage of Energetic Partices through Matter (CRC Press, Tayor Francis Group, New York, London, 2007). 6. N. Bohr, Phys. Rev. 58, 654 ( N. Bohr, Phys. Rev. 59, 279 (94). 8. P. Sigmund and U. Haagerup, Phys. Rev. A 34, 892 (986). 9. J. D. Jackson and R. L. McCarthy, Phys. Rev. B 6, 43 (972). 20. H. Bichse, Phys. Rev. A 4, 3642 ( S. P. Ahen, Rev. Mod. Phys. 52, 2 ( J. Heraut, R. Bimbot, H. Gauvin, et a., Nuc. Instrum. Methods Phys. Res. B 6, 56 (99). 23. R. Bimbot, C. Cabot, D. Gardes, et a., Nuc. Instrum. Methods Phys. Res. B 44, 9 (989). 24. H. Geisse, Y. Laichter, W. F. W. Schneider, and P. Armbruster, Phys. Lett. A 88, 26 (982). 25. H. Geisse, Y. Laichter, R. Abrecht, et a., Nuc. Instrum. Methods Phys. Res. B 206, 609 (983). Transated by R. Tyapaev JETP LETTERS Vo. 94 No. 20