Energy straggling of energetic light ion beams

Energy straggling of energetic light ion beams

Nuclear Instruments and Methods North-Holland, Amsterdam ENERGY STRAGGLING T. KANEKO and in Physics Research OF ENERGETIC B13 (1986) 119-122 ...

342KB Sizes 2 Downloads 129 Views

Nuclear Instruments and Methods North-Holland, Amsterdam

ENERGY

STRAGGLING

T. KANEKO

and

in Physics

Research

OF ENERGETIC

B13 (1986)

119-122

LIGHT ION BEAMS

Y. YAMAMURA

Department of Applied Physics, Okayama University of Science, Ridai-cho,

The energy straggling using two local electron treated. At low energies, the value predicted by straggling due to change atom% number)-oscillation.

119

Okayama 7G0, Japan

of protons and helium ions is investigated based on the dielectric function theory of Lindhard and Winther density models (LEDM). Not only point charges (H’, He”), but also a partially stripped ion (He’), are the collisional straggling of point charges is proportional to the energy, and at high energies it approaches Bohr. The solid LEDM calculation yields larger straggling values than the atomic LEDM does. The state fluctuations of a 250 keV/amu helium beam results in an enhancement of the well-known Zz (target

1. Introduction

In ion-beam material interactions, the energy loss of charged particles has received a great deal of attention for many years since it plays an important role in investigating the composition, the depth distribution, and the location of the lattice site of implanted atoms in the matter. The energy spread around the average energy loss, i.e. the energy straggling, limits directly and ultimately the spatial resolution of the implanted atom distribution. The energy straggling is caused by statistical fluctuations in the collision processes that particles are subjected to during the passage. Several theories are available to explain the so-called collision energy straggling due to the transfer of a squared energy to target electrons from the projectile. Following Bohr [l], the straggling of a particle with atomic number Z, and velocity u penetrating matter with atomic number Z, is given by Qi = 4nZ:Z,e4Nx,

(1)

in the high velocity region, where N and x denote the number density of target atoms and the path length of the projectile, respectively. In order to apply the above formula at low velocities, Lindhard and Scharff [2] extended Bohr’s formula by using a velocity-dependent factor. Refinements of the Lindhard-Scharff theory were performed by Bonderup and Hvelplund [3] using the Lenz-Jensen model for the target electron distribution. Chu [4] derived the cross-over feature for different target atoms at the same velocities for protons or helium ions using Hartree-Fock wave functions, resulting in the well-known Z, oscillation. Apart from the collisional straggling, the extra contributions coming from, e.g. target thickness variation, “bunching” effect, and charge state fluctuations, have also been discussed recently [5]. The aim of the paper is to evaluate the straggling of 0168-583X/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

energetic protons and helium ions (including a partially stripped ion) penetrating solid and gaseous media, where the charge state distribution in the matter is taken into account. The present calculation is based on the Lindhard-Winther (LW) theory together with local electron density models. The extra contribution of charge state fluctuation to the energy straggling is also estimated for a 250 keV/amu helium ion beam.

2. Basic description As energetic particles pass through matter, they lose their energy and change their charge state through successive collisions with the target atoms. Traversing a sufficiently long distance, the charge states of the particles reach the equilibrium state. Let us consider the particle transport including a variable denoting the charge state [6]. In the case of detecting the emergent particles in all charge states, which are all allowed in matter, the average of the square of the energy transfer, (E*), of the emergent ion beam differs from the square of the average energy transfer (E) by .n’=(E2)-(E)Z.

(2)

In our treatment, this energy straggling 0* consists of two parts. One is the collisional straggling stated in the introduction, which can be obtained for ion beams by averaging the single particle straggling in a particular charge state i, flf, over the charge state fractions, the 4i’s. That is to say. we obtain

Most experiments were performed under charge state equilibrium conditions, then the 4,‘s should be replaced by their equilibrated values. The points stressed here are to introduce via eq. (3) the charge state distributions and to treat the straggling of a partially III. STOPPING

POWER/ENERGY

LOSS

T. Kaneko,

120

Y. Yamamura

I Energy straggling

stripped ion. For partially stripped ions the collisional straggling has the form in atomic units of

of energetic light ion beams

the total electron number per atom in a solid. A more detailed description of the LEDM is given in refs. [9] and [lo].

R:=NXIU^dww’~~~“dk(2/~ku2) 3. Numerical x IZ, - p,(k)l’ Im(Ile(k,

w)) ,

(4)

which was obtained by extending the Lindhard-Winther theory [7]. In the above, ~(k, w) is the dielectric function of the medium and Im(. .) means the imaginary part. In addition, p,(k) denotes the Fourier transform of the spatial distribution of the electrons, p,(r), bound to the ion, which is assumed to be spherically symmetric. If we take the 1s type hydrogenie wave function #cl(r) = (~u~))“~ exp(-rla,)(a, = l/Z,) to express p,(r)(=]+(r)]*) of an He’ ion, then we obtain p,(k) = [l + (ka,/2)2]~*

(5)

It is well known that the imaginary

part of the inverse dielectric function is contributed from the electronhole pair excitation as well as the resonance or collective excitation. Apart from the stopping cross sections, the resonance excitation branch contributes negligibly in comparison with the pair excitation one [3]. The other contribution to the straggling comes from the charge state fluctuations. In the case in which two charge components are dominant and labelled as 0 and 1, this contribution is found [5] to be Q:, = Nx2(5,

- 5,)%,~,/(o,,,,,

+ oc,,,) 3

(6)

with 40 = oCcBP/(oCcaP + %,,)>

$1 = %X,~(o,,,

+ %J

7 (7)

where S,(S,) and &($,) denote the stopping cross section for and the charge state fraction of the particle in charge state l(O), respectively, and a,,, (g,,,S) is the electron-capture (electron-loss) cross section for the particle in charge state l(0). Except for the chargechanging cross sections, eq. (6) is governed by the difference of the stopping cross sections in two charge states. In order to calculate the straggling 0: and the stopping cross section Si (i = 0, l), we adopt local electron density models (LEDM) from a practical point of view. In the atomic LEDM, which serves to put into a more realistic form the spatial electron distribution of the neutral atoms, Hartree-Fock wave functions [8] are used to take into account the electronic shell structure. In the solid LEDM, the constant electron density is introduced in the outer region of the Wigner-Seitz cell for the purpose of including the collective motion of the conduction electrons, characterized by the bulk plasma frequency oP [9]. The inner region of the Wigner-Seitz cell is also described by Hartree-Fock wave functions so that the local density is normalized there to conserve

results and discussion

Fig. 1 shows the reduced energy straggling n/n,, coming from the collisional part, of a point charge passing through solid targets, i.e., Be, B, Mg, Al, Si, Zn, Ge, and Sb, where the number of the free electrons per atom are assumed to be two, three, two, three, four, two, four, and five, respectively. In the low keV/amu energy region, R2 is approximately proportional to E, corresponding to the result of the LW theory even when the local electron density model is adopted. For light elements, e.g., Be and B, the contribution from the core electrons causes about ten and twenty per cent of the total straggling, respectively. For heavy elements, however, this contribution becomes comparable with the contribution from the free electrons when E 5 50 keV/amu. With increasing energy, the core electrons are dominant and, in an extreme case, the contribution from the free electrons can be neglected, where the Bohr formula become valid and only the number of electrons is important to estimate R. In fig. 2, the well-known Z, oscillations in RZ of a proton with 100,200, and 400 keV are reproduced using the atomic LEDM. For comparison, the results using the solid LEDM are also plotted. They yield greater values than the atomic LEDM do. In addition, the phase of the Z,-oscillation is shifted a bit towards the high Z, direction with increasing energy. In fig. 3, the straggling of 250 keV/amu He ion beams versus Z, are illustrated, where the straggling of a 250 keV/amu He’+ is obtained by multiplying that of a H’ ion ion, a;,,+, with the same keV/amu energy by a factor 4. The straggling of a 250 keV/amu He’ ion, ah,+, is calculated with the use of a, = 0.5~~ (a, is the Bohr radius). The straggling ai,+ has a similar structure to 0&z+. The straggling of the helium ion beam, a;,, is obtained by averaging fl&+ and Oiez+ over observed charge state fractions (CSF) [ll], since at this energy the neutral component, He’, can be neglected. Here we note that even if we include the Z, oscillation in the average charge of the MeV helium ion beams into 0 i,, the Z, oscillation of ok, is compatible with that of either a:,, or 0he2+. The contribution from the ‘charge state fluctuation, 0:,, is estimated using consistently the observed CSF data and the theoretical loss cross sections. As the CSF data only provide us with the ratios of the electron-capture and -loss cross sections, we utilize here the theoretical calculation of the loss cross sections, based on the unitarized impact parameter method [12]. Then, the capture cross sections are determined straightforwardly from eq. (7). The stopping cross sections for a He’ and He’+, i.e., S,,, and

T. Kaneko,

Y. Yamamura

( MeV/amu

E

I Energy straggling of energetic light ion beams

121

)

1.0

( keV/amu

E Fig. 1. The reduced (LEDM) is used.

energy

103

102

10

straggling

n/R,

)

of a point

charge

in various

solids,

where

the solid local-electron-density-model

STRAGGLING 20

c

0

250

0

-0

keV/amu

A AA

l*

AA

v’ AA A~AAAAAA&A oo--=m~ 00 l

co

X” A A



0”

2

0

xx + ++

x

He’*

o

He*

A He beam(coll.)

Fig. 2. The calculated energy straggling of a proton 200, and 400 keV versus the target atomic number both the solid and the atomic LEDM’s.

with 100, Z, using

Fig. 3. The calculated energy straggling of a He”, a He’, the He beam with 250 keV/amu versus Z,: X; He*‘, 0; He’, A; He beam, 0; He beam including charge state fluctuation. III. STOPPING

POWER/ENERGY

LOSS

T. Kaneko,

122

Y. Yamamura

I Energy straggling of energetic light ion beam

S HeZ+, are also calculated using the LW theory and the LEDM [ll]. In comparison with the straggling data [13], the collisional straggling yields lower values, and therefore the inclusion of Of, improves this discrepancy a bit. In order to obtain a more satisfactory agreement, the other extra contributions will be investigated quantitatively as well as the charge-changing one for various energies and targets in the future. This work is financially supported by the Special Project Research on Ion Beam Interaction with Solids from the Ministry of Education, Science and Culture, Japan. In part, Institute of Plasma Physics at Nagoya University, Japan, also supports the work.

References [l] N. Bohr, K. Dan. Vid. Selsk. Mat.-Fys. Medd. 18 (1948)

No. 8.

[2] J. Lindhard and M. Scharff, K. Dan. Vid. Selsk. Mat.Fys. Medd. 27 (1953) No. 15. [3] E. Bonderup and P. Hvelplund, Phys. Rev. A4 (1971) 562. [4] W.K. Chu, Phys. Rev. Al3 (1976) 2057. [5] F. Besenbacher, J.U. Andersen and E. Bonderup, Nucl. Instr. and Meth. 168 (1980) 1. [6] T. Kaneko and Y. Yamamura, to be submitted. [7] J. Lindhard and A. Winther, K. Dan. Vid. Selsk. Mat.-Fys. Medd. 34 (1964) No. 4. [S] E. Clementi and C. Roetti, Atom. Data Nucl. Data Tables 14 (1974) 177. [9] I. Gertner, M. Meron and B. Rosner, Phys. Rev. A18 (1978) 2022; A21 (1980) 1191. [lo] T. Kaneko, Phys. Rev. A30 (1984) 1714; T. Kaneko, to be submitted. [ll] Y. Haruyama, Y. Kanamori, T. Kido and F. Fukuzawa, J. Phys. B15 (1982) 779; Y. Haruyama, Y. Kanamori, T. Kido, A. Itoh and F. Fukuzawa, J. Phys. 816 (1983) 1225. [12] T. Kaneko, Phys. Rev. A32 (1985) 2175. [13] For example, J.B. Malherbe and H.W. Alberts, Nucl. Instr. and Meth. 192 (1982) 559.