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NI M B
Beam Interactions with Materials & Atoms

Nuclear Instruments and Methods in Physics Research B 245 (2006) 28­31 www.elsevier.com/locate/nimb

Alignment dependence of the stopping effective charge of swift excited ions in the degenerate electron gas
L.L. Balashova *, A.A. Sokolik
Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russian Federation Available online 4 January 2006

Abstract The mation Li+ ion ñ 2005 dielectric theory of ion­matter interaction is extended to the case of excited swift ions to investigate possible influence of defor(alignment) of their electron cloud on their stopping. Calculations are performed for the stopping effective charge of fast lithium and neutral helium atom in carbon in the velocity range 5­20 a.u. Elsevier B.V. All rights reserved.

PACS: 34.50.Bw; 34.90.+q Keywords: Atomic collisions; Energy loss; Stopping power; Alignment

1. Introduction This work was undertaken as a contribution to general discussion of the role of exited states of swift ions in their stopping kinetics [1,2]. Recently, Tsuchida and Kaneko [3] introduced a concept of instant stopping power for an ion in fixed excited states of its electron cloud. Neglecting all charge-exchange and ion angular-momentum re-orientation processes due to ion­medium interactions they used their so-called frozen-ion approximation to put attention to possible influence of angular anisotropy of the electron cloud on ion stopping. Using the first Born approximation within the binary ion­atom approach they showed that the stopping power for helium-like ions in excited states 1s2p:1P1M depends on the distribution of magnetic quantum number M. We stay within the same schematic frozen-ion approach to the problem but, addressing our studies to the case of ion­plasma and ion­solid interactions, use the dielectric theory of stopping [4,5]. Besides, we combine this approach

with the general formalism of the theory of angularmomentum alignment effects in collisions [6]. This enables to investigate the problem not only when the ion is in one of its pure jLMi states but also in general case of ionic mixed states. As usually, the stopping effective charge qeff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for a frozen ion is defined as S ion =S proton . Stopping of helium atom and helium-like lithium ion Li+ in 1s2p:1P1 state is taken as an example for calculations. They are preceded by more general discussion on possible influence of anisotropic distribution of the ion electron cloud on the profile and magnitude of the polarization wake produced by the projectile in the electron gas. 2. The Tsuchida­Kaneko approach Stopping power þdE/dx = NaS(v) for a swift ion in a frozen-charge state i is calculated in [3] as Z kmax X dk 2 2 2 S i Ïv÷ ¼ ÏE k þ E i ÷ á 8pÏe2 = ÷ jF p Ï~÷j jF tki Ï~÷j . hv k ik k3 k min k Here, Na is the number of target atoms per unit volume; F p Ï~÷ and F tki Ï~÷ stand for the elastic scattering amplitude k ik (elastic form-factor) of the projectile and the inelastic scattering amplitude for the atom. After applying the Bethe

Corresponding author. E-mail addresses: balash@anna19.npi.msu.su (L.L. Balashova), aasokolik@yandex.ru (A.A. Sokolik). 0168-583X/$ - see front matter ñ 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.059

*


L.L. Balashova, A.A. Sokolik / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 28­31

29

sum rule the ion stopping power is expressed in the standard form: S i Ï v÷ ¼ 4 pe 2 N a Z 2 ½L0 Ïv÷× DLÏv÷. me v2

r rectly due to dependence of the wake potential V Ïwake÷ Ï~÷ on the electron density distribution qe Ï~÷. r We take the dielectric function Ï~; x÷ for the degenerate k electron gas according to the plasmon pole approximation [4]: x2 1 p þ1¼þ . Ïk ; x÷ x2 × b2 k 2 × k 4 =4 þ xÏx × i0÷ g The plasmon frequency xp = (4pne)1/2 and the effective band-gap energy xg in semiconductors and insulators form together the collective resonance frequency X0 ¼ 1=2 1=2 Ïx2 × x2 ÷ . Parameters ne and b ¼ Ï3 k 2 ÷ stand for the p g 5F average electron density of the medium and the hydrodynamic velocity of disturbances in the electron gas; k F ¼ 1=3 1=3 2 Ï3p2 ne ÷ ¼ Ï3 p÷ xp=3 is the Fermi momentum of elec4 trons. In isotropic medium the dielectric function Ï~; x÷ k does not depend of direction of vector ~. On the other k hand, contrary to usual applications of the dielectric theory, the orientation dependence of qe Ï~÷ on the wave vector k ~ is of principal importance for our further calculations. k Further discussion is devoted to a particular case of two-electron atoms and ions whose ground state is e3 e isotropic and described with qe Ï~÷ ¼ Ï2=p÷ Z 1 eþ2 Z 1 r where r 1 e Z 1 ¼ Z 1 þ 5=16. In case of excited 1s2p: P1 state we parameterize the electron density distribution i pffiffiffi 1h 2 R1s Ïr÷× R2 Ïr÷ 1 þ 2A20 Ï1; 1÷P 2 Ïcos hr ÷ ; qe Ï~÷ ¼ r 2p 4p Ï 4÷ using the alignment parameter A20(1, 1) which shows relative population ffiffiffi ion magnetic sublevels jLMi [6] p of and varies from þ 2 in the case of pure 1P1,M=0 state to 1 pffiffi in the opposite case of equally populated states 2 1 e e P1,M=±1. We take parameters Z Ï1s÷ ¼ Z 1 and Z Ï2p÷ ¼ Z 1 þ 1 for hydrogen-like radial wave functions R1s(r) and R2p(r) in Eq. (4); P2(cos hr) is the Legendre polynomial. In Fig. 1(a) and (b) we demonstrate our calculation for the longitudinal profile of the wake potential produced by

General conclusion of paper [3] on the angular-momentum orientation effect in ion stopping is formulated by equations: S
1s2pÏM ¼0÷


1s2pÏisotropic÷

%S

1s2pÏM ¼ô1÷

;

the relative difference at v P 10 a.u. is less than 7% between S1s2p(M=0) and S1s2p(isotropic) and less than 2% between S1s2p(M=±1) and S1s2p(isotropic). 3. Polarization wake produced by a frozen excited ion in the degenerate electron gas In the dielectric theory the ion is calculated as convolution of tribution qion Ï~; t÷ of the ion r ~Ï~; t÷ ¼ þrV Ïwake÷ Ï~; t÷ produced Er r through the medium: Z ~ v qion Ï~; t÷~Ï~; t÷ d~; r Er r S ion Ïv÷ ¼ þ v stopping effective charge the charge density disand the electric field by the ion on its passage

Ï1÷

r with the electron cloud density qe Ï~÷ as a part of the total ion charge density distribution r r vt r vt qion Ï~; t÷ ¼ Z 1 dÏ~ þ ~ ÷þ qe Ï~ þ ~ ÷.
Ïwake÷

Ï2÷

Ï~; t÷ pror Atomic units are used. Wake potential V duced by this charge flux reads [4] Z i~~ h i 1 e kr Ïwake÷ V Ï~÷ ¼ 2 r Z 1 þ qe Ï~÷ k 2 2p k ! 1 á à d~; k Ï3÷ þ1 Ï~; x÷ k x¼~v k~ R ~r k rr where qe Ï~÷ ¼ eþik~qe Ï~÷ d~. Eqs. (1)­(3) show that the concept of deformed electron cloud distribution of the moving ion enters integral (1) directly via (2) and also indi-

0. 4

(a)

0. 4

(b)

1s2p M= 0

(a.u.)

(a.u.)

1s
0. 0

2

v

0. 0

v

(wake)

V

V
-0 .4 -0 .4

(wake)

1s2 p
-0 .8 -50 -40 -30 -20 -10 0 10 -0 .8 -50 -40

1s2p M= ±1

-30

-20

-10

0

10

z (a.u.)

z (a.u.)

Fig. 1. The wake potential along the projectile trajectory for Li+ pffiffiffi at v = 5 a.u.: (a) in ground 1s2 (--) and isotropic excited 1s2p (A20(1, 1) = 0, ­ ­ ­) ion 1 states; (b) in aligned excited 1s2p state with M =0, A20 Ï1; 1÷ ¼ þ 2 (--) and M = ±1, A20 Ï1; 1÷ ¼ pffiffi. 2


30
0. 2

L.L. Balashova, A.A. Sokolik / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 28­31
2. 2 2. 1

1s2p isotropic: A 20 (1 ,1 )= 0

1s2p

M= ± 1

1s
0. 0

2

v

2. 0 1. 9

1s2p

M= 0

(wake)

(a.u.)

V

q
-0 .2
1. 8

ef f

1s2p

1. 7 1. 6

1s

2

1s 2p

M= ± 1

-0 .4 -50 -40 -30 -20 -10 0 10

1s 2p

M= 0

z (a.u.)

1. 5 0 5 10 15 20

Fig. 2. The same as Fig. 1(a) but for He neutral atom.

v (a.u.)
Fig. 3. Velocity dependence of the stopping effective charge qeff for Li+ ions propagating through carbon target.

ion Li+ propagating through carbon target (ne = 0.052 a.u.). Comparing the solid line results (the ground state 1s2) with those given by dashed line (the isotropic excited 1s2p state: A20(1, 1) = 0) shows that electron screening of ion nucleus charge is decreasing with ion excitation considerably. At the first one-fourth of the wake wave length behind the ion this effect is about 30%. Influence of the alignment (Fig. 1(b)) is also pronounced but is less than 10%. Similar tendencies can be seen in case of neutral helium atom (Fig. 2) where the effect of ion excitation is much stronger. 4. Stopping effective charge of excited helium atom and helium-like Li+ ion Using relation x ¼ ~v ¼ kv à cos hk , Eq. (1) transforms in k~ k-representation into Z Z1 i2 2 dk kv h 2 S ion ¼ 2 Z 1 þ qe Ïk ; Ïx=kv÷ ÷ pv 0 k 0 1 á Im þ x dx. Ïk ; x÷ The same formula, after changing the bracket [Z1 þ qe(k,(x/kv)2)] by Zp = 1, gives the stopping power Sp of proton. Their ratio determffiines the stopping effective pffiffiffiffiffiffiffiffiffiffiffiffiffi charge of the ion qeff ¼ S ion =S p . This relation reduces to that in usual case of ions with isotropic electron cloud [4­8] when one miss term (x/kv)2 in qe(k,(x/kv)2). Generally, alignment dependence of the stopping power is expected to be most appreciable at small values of the projectile atomic number Z1 due to the maximal role of the ion electron cloud in such cases. This is confirmed by numerical calculations for neutral He atom and Li+ ion, both in the ground and in the excited 1s2p:1P1 states. As an example in Fig. 3 the velocity dependence of the stopping effective charge is shown for Li+ ion in solid carbon. In the ion velocity range from 4 to 20 a.u. (Eion = 2.8­70 MeV) the stopping effective charge is increasing

gradually with energy under any form of the angular distribution of the electron cloud density. But in 1s2 case, contrary to 1s2p one, the effective charge approaches its asymptotic limit Z1 = 3 not so fast because of considerable screening of the nuclear charge by both 1s electrons. The difference caused by the alignment is about 7% only. When the electron cloud of the ion is prolate along its movement direction (1s2p:1P1,M=0 state) the stopping is smaller than for the isotropic electron cloud. In the opposite state of the oblate electron cloud (1s2p:1P1,M=±1 state) the stopping is greater although the deviation from the case of isotropic state is less. All these regularities are also well pronounced in the stopping effective charge of neutral helium atom. For higher Z1 the alignment effect is much smaller because of greater role of nucleus charge masking dependence of ion stopping characteristics on details of the ion electron cloud distributions. 5. Conclusion Using the dielectric theory of stopping, we performed a schematic analysis of possible influence of alignment of the electron cloud of moving frozen excited ions on the produced polarization wake and, as a consequence, on the ion effective stopping charge. In the latter case, our results confirm qualitative conclusions made earlier by Tsuchida and Kaneko [3] including their prediction of stronger stopping of oblate than of prolate ions. Quantitatively both paper [3] and our work show a small alignment effect. Taking into account strong electron loss processes accompanying excitation of ions in course of their passage through matter, it is too early to discuss a way to observe this effect experimentally. We consider our work as a step to shed light to the problem of stopping of excited ions in electron gas whose further theoretical consideration needs much more efforts unifying (perhaps, within a time-dependent density


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31

matrix approach) the ion excitation, alignment, disalignment and charge-exchange aspects of the problem. Acknowledgement The work was supported by the Russian Foundation for Basic Research (Grant 04-02-16742). References
[1] J.P. Rozet, C. Stephan, D. Vernhet, Nucl. Instr. and Meth. B 107 (1996) 67.

[2] V.V. Balashov, Nucl. Instr. and Meth. B 205 (2003) 813. [3] H. Tsuchida, T. Kaneko, J. Phys. B 30 (1997) 1747. [4] P.M. Echenique, R.H. Ritchie, W. Brandt, Phys. Rev. B 20 (1979) 2567. [5] W. Brandt, M. Kitagawa, Phys. Rev. B 25 (1982) 5631. [6] V.V. Balashov, A.N. Grum-Grzhimailo, N.M. Kabachnik, Polarization and Correlation Phenomena in Atomic Collisions, Kluwer Academic, New York, 2000. [7] L.L. Balashova, N.M. Kabachnik, Izvestia Russ. Acad. Sci., Phys. 62 (4) (1998) 763. [8] Q. Yang, Phys. Rev. A 49 (1994) 1089.