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Pressure effects on the stopping and range of heavy ions
Nuclear Instruments and Methods in Physics Research B 222 (2004) 411–420 www.elsevier.com/locate/nimb
Pressure effects on the stopping and range of heavy ions q Salvador A. Cruz
*
Departamento de Fısica, Universidad Autonoma Metropolitana – Iztapalapa, Av. San Rafael Atlixco 186, Aparatado Postal 55 534, 09340 Mexico, DF, Mexico Received 29 February 2004
Abstract Pressure effects on the energy loss of heavy ions in compound target materials are analyzed through the cores and bonds (CAB) implementation of the kinetic theory of stopping for protons and the effective charge approximation. The effect of pressure on the target material is estimated through a model of molecular confinement whereby the changes in the molecular electronic properties are calculated selfconsistently. As pressure increases, a competing behavior between target density increase and reduction in the electronic stopping cross section is observed. As a consequence, total path ranges are decreased as compared with the corresponding pressure-free target. Range calculations are shown for Heþ (E 6 1:6 MeV) and Liþ (E 6 2:6 MeV) projectiles on dense water and methane targets. 2004 Elsevier B.V. All rights reserved. PACS: 31.90.+s; 34.50.Bw; 62.50.+P; 61.80.Az Keywords: Stopping power; Ion ranges; High pressures; Molecular confinement; Dense water; Dense methane
1. Introduction The study of ion penetration phenomena in dilute and condensed matter has been largely developed for more than 90 years. Both experiment and theory have formed an important feedback binomial for understanding the mechanisms through which a swift ion loses energy while traversing a given target material. From the theoretical point of view, even in the simplest case of a binary collision between an ion and an isolated target atom or molecule, the problem has a many-body character q Paper first read at the First International Symposium on Radiation Physics held in Mexico City, December 2003. * Tel.: +52-55-5804-4988; fax: +52-55-5804-4611. E-mail address: [email protected] (S.A. Cruz).
due to the interplay of crossed interactions among projectile and target electrons and nuclei. Furthermore, this complexity is increased in the case of a target material in its condensed phase, where non-local interactions induce a collective response of the medium as the projectile penetrates. The magnitude of the problem has demanded of the development of appropriate theoretical models whereby the relevant mechanisms of energy loss are judiciously introduced so that their relative participation in the stopping process allows understanding the experimentally observed behavior. In this spirit, experiment has played a paramount role to shed light on the dominant mechanisms of energy loss as a function of projectile/target type and energy. Moreover, the advent of more powerful and accurate experimental techniques has imposed
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.063
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further refinements to the theory in order to understand more subtle – yet important – details of the energy loss phenomenon such as the role of electron capture and loss mechanisms, band-gap effects in solid-state materials, physical phase-state effects and chemical binding effects to mention a few. Even though the theory of ion stopping has been developed with relative success in predicting range profiles and damage distribution for a wide class of materials leading to important applications in ion implantation techniques, still the above mentioned fundamental questions remain open for research. The purpose of this contribution is to present preliminary results on the behavior of the electronic stopping cross section (Se ) and total range of heavy ions penetrating a molecular target under high pressure. The motivation of this study has its root in recent work developed in our group [1–3] to treat consistently the effect of increasing target density (pressure) on chemical bond contributions to Se for protons. Indeed, if target density increases, one would expect that the electronic stopping cross section should be reduced according to its definition: Se ¼
1 dE ; n dx
ð1Þ
where n is the target number density and dE the dx electronic stopping power. It is not true, however, that the electronic stopping power for a target under pressure will remain the same as for the free target since this quantity is strongly dependent on the properties of its electronic structure. For a target under high pressures its electronic properties change strongly [2,3], hence a competing behavior between stopping power and target number density will yield a corresponding change in Eq. (1). In this work, the results obtained for proton stopping in dense molecular targets [2] are used to construct the corresponding electronic stopping cross section of heavy ions (SeHI ) using – as a first approximation – conventional effective charge scaling theory [4] whereby 2
SeHI ðvÞ ¼ ðcZHI Þ Sep ðvÞ;
ð2Þ
where c is the fractional effective charge term (see further below), ZHI the ion nuclear charge and Sep
the stopping cross section for protons moving with the same velocity ðvÞ as the heavy ion in a given medium. The total ion range is then estimated as a function of target pressure and compared with its corresponding value at normal pressure. Taking liquid water and methane under high pressure as sample media and Heþ (E 6 1:6 MeV) and Liþ (E 6 2:6 MeV) as projectiles, the associated pressure-dependent range-energy curves are constructed. It is found that the total range is effectively reduced as target pressure increases due the combined effect of target density increase and a corresponding decrease in SeHI . The work is organized as follows. In Section 2, a brief description of the theoretical assumptions and development for the treatment of molecular stopping under pressure is presented along with the results and a discussion of the main findings. In Section 3 the main conclusions of this study are drawn.
2. Theoretical considerations Consider a given molecular target in its condensed phase and subject to high pressure. For the purposes of our discussion, we shall assume here the simplest approach to a medium in its condensed phase, i.e. no account of phase diagramgoverned transitions and an equation of state (EOS) equally valid for the solid and the liquid phase. In this context, we consider the medium as formed by closely packed molecular entities each one of them spatially confined by its neighbors. Furthermore, we assume that the intermolecular potential is such that an infinitely high barrier potential appears at the boundary of the confinement volume. Accordingly, as pressure is increased the confinement volume for each molecule is reduced with consequent changes in the molecular electronic properties [1–3,5]. Since the inelastic energy loss mechanisms for an incoming projectile are inherently related to the projectile-target electronic structure, our goal is then to incorporate the changes in molecular electronic properties in the governing stopping power equation. Hence, a first step on call is to account for changes in the molecular properties as a function of pressure.
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
2.1. Model of molecular confinement According to our previous discussion, we shall consider a molecule in a dense medium as a system confined within an impenetrable spherical cavity whereby changes in pressure correspond to changes in cavity volume. In recent work we have discussed in detail the ab initio treatment of molecules confined by hard-walled spherical cages [3]. Here we give a brief summary of the main assumptions and results which are useful for the purposes of this work. Fig. 1 shows schematically a CH4 molecule within a spherical cavity of radius R. Clearly, the first change we may foresee for the confined molecule is a redistribution of its electronic density within the cavity. Indeed, this idea was first explored for the water molecule by Xu et al. [6] in their analysis of physical-phase state on proton stopping. These authors considered the electronic charge redistribution by renormalizing the freemolecule charge density within the confinement volume. However, the renormalized density is not consistent with the actual dynamical changes taking place for the molecule to attain its final stable conformation. Once the molecule is confined, the
R
r
r – Rk
electronic charge redistributes with consequences on the nuclear positions and molecular energy. This problem has been addressed [1–3,5] by treating the molecular ground state energy within an ab initio scheme using the floating spherical Gaussian orbital (FSGO) representation of localized molecular orbitals [7] modified such that for a confined molecule each electron pair in a given core, bond or lone-pair orbital is represented as [8] 2
Wk ðr Rk Þ ¼ Nk eak ðrRk Þ ð1 r2 =R2 Þ;
Fig. 1. Schematic diagram showing a methane molecule enclosed in a spherical cavity of radius R. The core and bond FSGO orbitals are shown as spheres of radius qk ¼ a1=2 , where k ak is the orbital parameter (see text). The orbital center positions Rk are also shown.
ð3Þ
where Nk is the normalizing constant, 1=2 ak R2 k
Nk ¼ ð4pÞ Z
e
R
2
2
e2ak r i0 ð4ak Rk rÞð1 r2 =R2 Þ r2 dr
1=2
0
ð4Þ and Rk the position of the center of orbital k relative to the origin and r the position of any point in space relative to the origin within the confinement space (see Fig. 1). ak in Eqs. (3) and (4) is an orbital parameter which, together with Rk , are determined after minimization of the total molecular energy for a given box radius. Note that the term in parenthesis in Eq. (3) is a cut-off term which guarantees that the wavefunction vanishes at the boundary. Hence, the energy minimization procedure described provides the equilibrium charge density and geometric molecular conformation for the cavity radius considered. Table 1 displays, as an example, the behavior of the ground state energy of the water and methane molecules as a function of confinement radius and pressure [2,3]. Pressure was obtained from an interpolated curve for the E versus R values as
ρk = α k – 1 / 2 Rk
413
P ¼
oE 1 oE ¼ : oV 4pR2 oR
ð5Þ
Fig. 2 shows the behavior of the orbital parameter ak and bond length as a function of cavity radius for the C–H and O–H bonds of methane and water, respectively. Since the FSGO orbital radius 1=2 is given as qk ¼ ak (from the argument in the exponential in Eq. (3)), from Fig. 2 we note that the orbitals become more compact as the cavity shrinks. Furthermore, for this isotropic compression, the bond length is reduced and – in the case
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Table 1 Ground state energy and associated pressure for the confined H2 O and CH4 molecules as a function of cavity radius calculated through the FSGO method adapted for confined systems [2,3] H2 O
CH4
R (a.u.)
Total energy (Hartrees)
0.85 1.0 1.25 1.5 1.75 2.0 2.5 3.0 4.0 5.0 6.0 10.0 1
Pressure (GPa) 5
6.8 · 10 2.9 · 105 8.3 · 104 2.8 · 104 1.1 · 104 4.4 · 103 940 254 28.8 4.9 1.1 0.015 104
1.983 )22.628 )43.390 )53.119 )58.097 )60.782 )63.114 )63.862 )64.200 )64.258 )64.275 )64.286 )64.287
orbital radius (a.u.)
α
-1/2 C-H
1.4 1.2
α
0.8 (a)
bond length (a.u.)
0.6
2
C-H
1.5
O-H
1 0.5 (b) 4
6 R (a.u.)
7.1 · 105 3.2 · 105 1.0 · 105 3.7 · 104 1.5 · 104 7.0 · 103 1.7 · 103 532 75.6 15.6 4.2 0.09 104
2.2. Proton stopping cross section in a dense medium
1
2
80.042 20.659 )4.114 )16.356 )23.075 )27.027 )31.011 )32.680 )33.727 )33.924 )33.967 )33.990 )33.992
-1/2
O-H
0
Pressure (GPa)
of water – the bond angle increases while for methane the bond angle remains the same [3]. Hence, molecular confinement not only affects the electron distribution but the whole molecular characteristics. This behavior will certainly have important consequences on the electronic stopping cross section as we discuss below.
1.8 1.6
Total energy (Hartrees)
8
10
Fig. 2. Behavior of (a) the orbital radius qk ¼ ak1=2 and (b) C– H and O–H bond lengths for water and methane as the confining radius is reduced [3].
We shall consider now the main elements leading to the calculation of the electronic stopping cross section for protons (Sep ) moving in a medium under high pressure. To this end, we resort to the orbital implementation of the kinetic theory of stopping [9–11]. An advantage of using this procedure is that we may evaluate the cores and bonds (CAB) contributions to Sep and directly take into account deviations from Bragg’s additivity rule [2,12,13]. In the kinetic theory of stopping [14] the electrons of the medium are not considered at rest but moving with a velocity distribution f ðv2 Þ, where v2 is the electron velocity relative to the laboratory system. In the frame of reference moving with a projectile of charge Z1 and non-relativistic laboratory velocity v1 , the relative velocity of the target scatterers is thus v0 ¼ v1 v2 . Hence, according to the kinetic theory of stopping the electronic stopping cross section, Se ðv1 Þ, may be written as [15]
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
Se ðv1 Þ ¼
Z
f ðv1 v0 Þ
v1 v0 S0 ðv0 Þ d3 v0 ; v 1 v0
ð6Þ
where S0 ðv0 Þ is the value for Se in the case v1 v2 (scatterers at rest). Making use of the orbital decomposition scheme proposed by Oddershede and Sabin [9–11]: X Se ðv1 Þ ¼ Se;k ðv1 Þ; ð7Þ k
where Se;k is the contribution from orbital ‘‘k’’, Se;k ðv1 Þ ¼
4pe4 Z12 Z2 Lk ðv1 Þ; mv21
ð8Þ
with m and e the electron mass and charge, respectively. Z2 is the total number of electrons of the molecular target and Lk the orbital stopping number, which in terms of the orbital mean excitation energy (Ik ) and its electron population (xk ) is given as [15] Z 1 2mv02 2 1 Lk ðv1 Þ ¼ pv1 Z2 xk ln dv0 Ik ak Z p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fk v21 þ v02 2v1 v0 cos h dðsin2 hÞ: 0
ð9Þ
quired in Eqs. (9) and (10) are directly dependent on the electron charge density, which may be constructed after Eqs. (3) and (4) once the molecular parameters have been optimized for specific confinement conditions as described in the previous subsection. The detailed calculations leading to explicit expressions for fk and Ik have been reported elsewhere [2]. For completeness and without digressing too much from the aims of this paper, here we give a summary of the main findings in [2]. Accordingly, fk and Ik have been shown to be X 2 2 fk ðkÞ ¼ ð2lþ1Þ Nk2 e2ak Rk p l Z R 2 ak r2 2 2 2 e il ð2ak Rk rÞjl ðkrÞð1 r =R Þr dr ; 0
ð11Þ where jl ðyÞ and il ðzÞ are the lth-order spherical Bessel function and modified spherical Bessel function p offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the first kind, respectively, and k ¼ ðm=hÞ v21 þ v02 2v1 v0 cos #, according to Eq. (9). Also Z R 2 2 2 e2ak r ð1 r2 =R2 Þ i0 ln Ik ¼ 2pNk2 e2ak Rk 0
1=2
Here ak ¼ ðIk =2mÞ and fk ðjv1 v0 jÞ ¼ fk ðjv2 jÞ is the velocity distribution of the scatterers in orbital‘‘k’’ relative to the projectile. We have used the Bethe term for S0 in Eq. (6). The orbital excitation energies, Ik , may be evaluated through the orbital local plasma approximation OLPA [16,17] as 1=2 Z 1 4pe2 h2 qðrÞ ln Ik ¼ q ðrÞ ln d3 r; ð10Þ xk C k m where qk ðrÞ is the local electronic density in configuration space and qðrÞ is the local total electron density obtained after taking the angular average of each orbital charge density [17]. vk corresponds to the scaling parameter originally proposed by Lindhard and Scharff [18] to account for polarization effects. We set vk ¼ 1 in our case. Note that C in Eq. (10) indicates integration within the confinement volume only. Clearly, the orbital velocity distribution fk as well as the orbital mean excitation energy Ik re-
415
ð4ak Rk rÞ ln½4pe2 h2 qðrÞ=m r2 dr;
ð12Þ
with the angularly averaged total local electron density qðrÞ given as qðrÞ ¼ 2ð1 r2 =R2 Þ2 X 2 2 2 Nj Nk eAjk Rjk ajk Kjk eajk r i0 ð2ajk Kjk ÞTjk ; j;k
ð13Þ where Tjk are the elements of the inverse overlap matrix [2,7] and the following definitions hold: ajk ¼ aj þ ak ; Rjk ¼ jRj Rk j;
Ajk ¼ aj ak =ajk ; Kjk ¼ jaj Rj þ ak Rk j=ajk :
ð14Þ
The structure of Eqs. (12) and (13) suggests an important dependence of the mean orbital excitation energy on the confinement radius (hence on pressure, according to Eq. (5)). The corresponding confinement effects on the total molecular mean ionization energy may be deemed using the orbital
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partition rule suggested by Oddershede and Sabin [10]: " cores bonds X 1 X ln Imolecule ¼ xi ln Iicore þ xj ln Ijbond N i j # lone-pairs X lone-pair þ xk ln Ik ; ð15Þ k
where N is the total number of electrons and xk the electron population of each core, bond and lone-pair orbital. Table 2 shows the orbital and total mean excitation energies for the water and methane molecules [2] as a function of pressure and the associated confinement radius. Clearly, as pressure increases, the mean excitation energy increases, as expected. This behavior brings as a consequence a corresponding reduction in the electronic stopping cross section as may be gathered after inspection of Eqs. (8) and (9). Fig. 3 shows, as an example, the predicted values for Sep as a function of energy for protons traversing water in its vapor phase (continuous curve) and for water under a 4.9 GPa pressure (dashed curve) [2]. The symbols in this figure correspond to experimental data for water in its gaseous phase [19] (solid triangles) and in its solid phase [20] (open circles). The well-known physical phase state effect is apparent in these experimental curves showing a lower value (10%) in Sep for the
condensed phase as compared to the gaseous phase. It is then reasonable to expect an increased lowering in the stopping curve for the high pressure case, as shown here. Fig. 4 shows the corresponding effect for protons incident on methane for various pressures as compared with the gaseous phase [2], where this effect becomes evident indicating also a small shift of the maximum of the sopping curve towards higher energies as pressure increases. 2.3. Scaling to heavy ion stopping Once Sep is obtained, the heavy ion stopping cross section (SeHI ) may be estimated using the effective charge scaling expression given by Eq. (2). In this work, the fractional effective charge (c) is calculated using the recommended expressions given by Ziegler et al. [4,21], based on the Brandt– Kitagawa theory [22]: 2 " 2 # 1 v0 K vF c ¼ q þ ð1 qÞ ln 1 þ 2:084 ; 2 a0 v 0 vF ð16Þ where q is the fractional charge state of the ion, v0 is the Bohr velocity, a0 the Bohr radius and vF the Fermi velocity of the medium, which is assumed as vF ¼ v0 in the case of organic compounds [21]. K in
Table 2 Core, bond, lone-pair and total mean excitation energies for the confined H2 O and CH4 molecules obtained from Eq. (12) for different confinement radii [2] H2 O
CH4
R (a.u.)
Icore
Ilone-pair
Ibond
Imolecule
Icore
Ibond
Imolecule
0.75 1.0 1.25 1.5 2.0 2.5 3.0 4.0 6.0 8.0 1
529.6 484.0 447.3 430.4 411.9 404.1 400.8 399.3 399.1 399.0 399.0
302.1 227.3 159.8 127.3 88.8 71.2 63.1 58.6 57.7 57.6 57.6
301.1 225.7 157.8 125.0 85.9 67.7 58.9 53.7 52.6 52.4 52.4
357.5 263.2 195.3 161.2 119.6 98.8 88.9 83.1 81.8 81.7 81.6
464.4 359.1 311.3 291.3 270.2 260.3 254.9 250.4 249.0 248.9 248.9
336.9 224.7 153.8 118.8 75.8 54.6 43.0 32.3 28.5 28.2 28.1
359.2 341.2 177.1 142.1 97.8 74.6 61.4 48.7 43.9 43.6 43.5
All energies are given in eV.
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q ¼ 1 exp½ð0:803yr0:3 þ 1:3167yr0:6
30 H → H2O
þ 0:38157yr þ 0:008983yr2 Þ ;
25 Free 20
Se(10
-15
P=4.9 GPa
10 5 0
50
100
150 200 250 Energy (keV)
300
350
400
H+ → CH4 40 gas phase 15.7 GPa 75.6 GPa
with ai ¼ ð0:2865; 0:1266; 0:001429; 0:02402; 0:001135; 0:00475; i ¼ 0; . . . ; 5Þ. Fig. 5 shows the theoretical stopping curve of Heþ ions incident on water at normal pressure (continuous curve) and at P ¼ 4:9 GPa (long-dashed curve) as a function of projectile energy. Also shown are available experimental data for the gaseous phase (solid circles [23] and open triangles [24]) as well as for the solid (open circles [23]) and liquid (inverted solid triangles [24]) phases. The short-dashed curve corresponds to scaled values obtained from the gas-phase theoretical curve after
532 GPa 80
0
2
4
v/v0
6
8
10
Fig. 4. Predicted pressure effects on the proton electronic stopping cross section for a methane target under various high pressure conditions [2]. The experimental data correspond to the gas phase: solid diamonds [19], open triangles [32], open diamonds [33].
-15
0
ev cm2/ molecule)
10
Se(10
Se(10
-15
ev cm2/ molecule)
50
20
with yr ¼ vr being the relative velocity of the ion’s electrons to the Fermi velocity of the medium approximated in this case (for v > v0 ) as vr vð1 þ 1=5v20 =v2 Þ. For Heþ projectiles, Ziegler et al. [4] suggest the use of a corrected fractional effective charge (cHe ) obtained after a systematic analysis of experimental data, namely " # 5 X i 2 cHe ¼ 1 exp ai lnðEÞ ; ð19Þ i¼0
Fig. 3. Electronic stopping cross section versus energy curve for protons incident on water under a 4.9 GPa pressure (dashed curve). Continuous curve: theoretical prediction for the gasphase at normal pressure. Experimental data correspond to the gas (solid triangles [19]) and solid (open circles [20]) phases.
30
ð18Þ
2=3 ZHI vr =v0 ,
2
ev cm / molecule)
+
15
417
2=3
K ¼ 0:686a0
1=3
ð1 qÞ ZHI : 1 17 ð1 qÞ
ð17Þ
The fractional charge state of an ion is given by Ziegler et al. as [4]
+
H 2O
60 gas phase
50
"scaled to condensed phase"
40
4.9 GPa
30 20 10
Eq. (16) is the dynamical screening length defined in the Brandt–Kitagawa theory in terms of the ion nuclear charge ZHI as
He
70
0
1
2
3 4 Energy (MeV)
5
6
7
Fig. 5. Theoretical predictions for the stopping cross section of Heþ ions incident on water vapor at normal pressure (continuous curve) and for water under a 4.9 GPa pressure (long-dashed curve). The short-dashed curve corresponds to the gas-phase curve reduced in 10% to account for the condensed phase (see text). Also shown are experimental points for water: gas phase (solid circles, [23]; open triangles, [24]); solid phase (open circles, [23]); liquid phase (inverted solid triangles, [24]).
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S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
reducing its amplitude in 10% to represent the observed physical phase-state difference. This latter average curve will be used when comparing total ranges within the same physical phase-state in water. In this preliminary study we shall consider only the energy region 0 6 E ðMeVÞ 6 1:6 and 0 6 E ðMeVÞ 6 2:6 for for Heþ and Liþ projectiles, respectively. As a matter of fact, in the above energy regions the stopping curves allow for the simple representation: SeHe ¼ Kua ebu
ðu ¼ v=v0 Þ;
ð20Þ
where the parameters K, a and b are determined after a best fit procedure to each curve. Table 3 displays the values of these parameters for the projectile-target combinations and pressures considered here. Eq. (20) will be useful for the total path range calculations given further below. 2.4. Total ion path range calculation The total ion path length has been calculated using the well-known expression for total stopping power:
dE ¼ nðSe þ Sn Þ; dx
ð21Þ
with Sn the nuclear stopping cross section given by Ziegler et al. [4]. In terms of the reduced velocity u ¼ v=v0 , the above expression may be cast as u
du ¼ ½Se ðuÞ þ Sn ðuÞ =k; dx
ð22Þ
with k¼2
Mp ðIH =nÞ; me
ð23Þ
where Mp , me , IH and n are the projectile atomic mass, the electron mass, Rydberg energy and target number density, respectively. From Eq. (22) the total range becomes then Z 0 u du R ¼ k ; ð24Þ u0 Se ðuÞ þ Sn ðuÞ with u0 the initial ion velocity. Using the fitted theoretical Se values (Eq. (20)) for Heþ (E 6 1:6 MeV) and Liþ (E 6 2:6 MeV) projectiles incident on water and methane as well as the corresponding Sn estimates according to [4], the integral in Eq. (24) was numerically evaluated. Note, however, that the factor k in Eq. (24) involves a reciprocal dependence on the target molecule number density n, as indicated by Eq. (23). For water and methane under high pressure, n was estimated from the respective pressure-density equation of state (EOS) of the compressed solid [25–27] using the pressure values obtained with our confinement model. Accordingly, for water at 4.9 GPa, q ¼ 1:64 g cm3 ; n ¼ 5:68 1022 molec cm3 while for methane at 15.7 GPa: q ¼ 1:09 g cm3 ; n ¼ 4:24 1022 molec cm3 and at 75.6 GPa: q ¼ 1:66 g cm3 ; n ¼ 6:49 1022 molec cm3 . Figs. 6 and 7 show the calculated pressuredependence of the total-path range versus energy curves of Heþ and Liþ projectiles, respectively, in
Table 3 Values of the parameters required in Eq. (20) for the analytical evaluation of the calculated electronic stopping cross sections for helium and lithium projectiles incident on water and methane at high pressures (see text) H2 O (‘‘liquid’’)
CH4
Projectile
Pressure (GPa)
K
a
b
Heþ Liþ
104
85.592 102.288
1.597 1.724
0.667 0.629
K
a
b
Heþ Liþ
4.9
53.255 64.489
1.944 2.135
0.693 0.682
Heþ Liþ
15.7
127.152 153.121
1.606 1.731
0.743 0.708
Heþ Liþ
75.6
108.882 132.495
1.651 1.804
0.725 0.704
K is in units of 1015 eV cm2 /molecule.
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
maximum. This behavior might reflect the competing role between target density increase and reduction in stopping cross section due to pressure increase as suggested by Eq. (24). A more detailed analysis is necessary including other projectiletarget systems to assess a definite conclusion.
10 +
Projectile: He
Total Range (µm)
8
1 atm
H O ( "liquid" )
6
419
2
4.9 GPa
4 CH
3. Conclusions
15.7 GPa
4
2
75.6 GPa
0 0.2
0
0.4
0.6
0.8
1
1.2
1.4
1.6
Energy (Mev)
Fig. 6. Predicted total path range-energy curves for Heþ ions traversing water and methane under various pressure conditions.
10 Projectile: Li
+
1 atm
Total Range (µm)
8 H O ( "liquid" ) 2
6
4.9 GPa
4
CH
15.7GPa
4
2
75.6 GPa
0 0
0.5
1.5 1 Energy (MeV)
2
2.5
Target pressure effects on the electronic stopping cross section and ranges of heavy ions have been analyzed in terms of effective charge theory and a molecular confinement model to simulate the effect of pressure. The results of this work are preliminary and predict an important reduction of the stopping cross section and range of swift ions traversing a target under high pressure. While the differences are more noticeable for high ion energies and gigapascal pressures, it is deemed here that a possible scenario where these differences might be observed is the length of fission fragment tracks formed in materials under high pressure as compared to those formed in the same material at normal pressure [28]. Indeed, recent experiments on fission track length registration and annealing in apatite under high pressure seem to give evidence of this effect [29], which has been pointed out – with some debate – as crucial for thermochronology studies [30,31]. Further experimental and theoretical work is necessary to ascertain the general validity of the results presented here.
Fig. 7. Same as Fig. 6 for Liþ ions.
Acknowledgements water and methane. For Heþ on water, the ‘‘liquid’’ phase results at normal pressure were obtained through the scaled gas-phase Se values mentioned at the end of the previous section. From these figures we observe a systematic reduction in the range values as pressure increases. This reduction may be considerably large mainly for higher energies. Note, however, a slight crossover in the range values for low projectile energies (EðHeÞ 6 0:4 MeV; EðLiÞ 6 0:6 MeV) in the case of water in its condensed phase at P ¼ 1 atm and P ¼ 4:9 GPa. This effect may be related with the steep reduction in the corresponding Se versus E curves below their
I am deeply indebted to Lew Chadderton for bringing to my attention the problem of pressure effects on the range of swift ions. I also appreciate the illuminating and inspiring discussions with him during his attendance to the First International Radiation Effects Meeting in Mexico City which certainly enriched the contents of this contribution. References [1] S.A. Cruz, J. Soullard, E.G. Gamaly, Phys. Rev. A 60 (1999) 2207.
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[2] S.A. Cruz, J. Soullard, Adv. Quantum Chem. 46 (2004) 195. [3] S.A. Cruz, J. Soullard, in preparation. [4] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [5] S.A. Cruz, Rev. Mex. Fis. 46 (2000) 207. [6] Y.J. Xu, G.S. Khandewal, J.W. Wilson, Phys. Rev. A 32 (1985) 629. [7] A.A. Frost, Theor. Chim. Acta 18 (1970) 156, and references therein. [8] S.A. Cruz, J. Soullard, Int. J. Quantum Chem. 83 (2001) 271. [9] J.R. Sabin, J. Oddershede, Phys. Rev. A 26 (1982) 3209. [10] J. Oddershede, J.R. Sabin, Nucl. Instr. and Meth. B 42 (1989) 7. [11] J. Oddershede, J.R. Sabin, Int. J. Quantum Chem.: Quant. Chem. Symp. 23 (1989) 557. [12] W.H. Bragg, R. Kleeman, Philos. Mag. 10 (1905) 318. [13] D.I. Thwaites, Nucl. Instr. and Meth. B 12 (1985) 84. [14] P. Sigmund, Phys. Rev. A 26 (1982) 2497. [15] R. Cabrera-Trujillo, S.A. Cruz, J. Soullard, Nucl. Instr. and Meth. B 93 (1994) 166. [16] D.E. Meltzer, J.R. Sabin, S.B. Trickey, Phys. Rev. A 1 (1990) 220, Phys. Rev. A 42 (1990) 666(E). [17] D.E. Meltzer, J.R. Sabin, S.B. Trickey, J.Z. Wu, Nucl. Instr. and Meth. B 82 (1993) 493. [18] J. Lindhard, M. Scharff, K. Dan. Videnskab. Selsk. MatFys. Medd. 27 (15) (1953).
[19] H.K. Reynolds, D.N.F. Dunbar, W.A. Wentzel, W. Whaling, Phys. Rev. 92 (1953) 742. [20] W.A. Wenzel, W. Whaling, Phys. Rev. 87 (1952) 499. [21] J.F. Ziegler, J.M. Manoyan, Nucl. Instr. and Meth. B 35 (1988) 215. [22] W. Brandt, M. Kitagawa, Phys. Rev. B 25 (1982) 5631. [23] R. Matteson, D. Powers, E.L.K. Chau, Phys. Rev. A 15 (1977) 856. [24] R.B.J. Palmer, A. Akhavan-Rezayat, J. Phys. D: Appl. Phys. 11 (1978) 605. [25] P. Loubeyre, R. LeToullec, E. Wolanin, M. Hanfland, D. Hausermann, Nature 397 (1999) 503. [26] S. Umemoto, T. Yoshii, Y. Akahama, H. Kawamura, J. Phys.: Condens. Matter 14 (2002) 10675. [27] F. Ancilotto, G.L. Chiariotti, S. Scandolo, E. Tosatti, Science 275 (1997) 1288. [28] L.T. Chadderton, private communication. [29] A.S. Wendt, O. Vidal, L.T. Chadderton, Earth Planet Sci. Lett. 201 (2002) 593. [30] B.P. Kohn, D.X. Belton, R.W. Brown, A.J.W. Geadow, P.F. Green, J.F. Lovering, Earth Planet Sci. Lett. 215 (2003) 299. [31] O. Vidal, A.S. Wendt, L.T. Chadderton, Earth Planet Sci. Lett. 215 (2003) 307. [32] H. Bichsel, L.E. Porter, Phys. Rev. A 25 (1982) 2499. [33] J.T. Park, E.J. Zimmerman, Phys. Rev. 131 (1963) 1611.
Pressure effects on the stopping and range of heavy ions q Salvador A. Cruz
*
Departamento de Fısica, Universidad Autonoma Metropolitana – Iztapalapa, Av. San Rafael Atlixco 186, Aparatado Postal 55 534, 09340 Mexico, DF, Mexico Received 29 February 2004
Abstract Pressure effects on the energy loss of heavy ions in compound target materials are analyzed through the cores and bonds (CAB) implementation of the kinetic theory of stopping for protons and the effective charge approximation. The effect of pressure on the target material is estimated through a model of molecular confinement whereby the changes in the molecular electronic properties are calculated selfconsistently. As pressure increases, a competing behavior between target density increase and reduction in the electronic stopping cross section is observed. As a consequence, total path ranges are decreased as compared with the corresponding pressure-free target. Range calculations are shown for Heþ (E 6 1:6 MeV) and Liþ (E 6 2:6 MeV) projectiles on dense water and methane targets. 2004 Elsevier B.V. All rights reserved. PACS: 31.90.+s; 34.50.Bw; 62.50.+P; 61.80.Az Keywords: Stopping power; Ion ranges; High pressures; Molecular confinement; Dense water; Dense methane
1. Introduction The study of ion penetration phenomena in dilute and condensed matter has been largely developed for more than 90 years. Both experiment and theory have formed an important feedback binomial for understanding the mechanisms through which a swift ion loses energy while traversing a given target material. From the theoretical point of view, even in the simplest case of a binary collision between an ion and an isolated target atom or molecule, the problem has a many-body character q Paper first read at the First International Symposium on Radiation Physics held in Mexico City, December 2003. * Tel.: +52-55-5804-4988; fax: +52-55-5804-4611. E-mail address: [email protected] (S.A. Cruz).
due to the interplay of crossed interactions among projectile and target electrons and nuclei. Furthermore, this complexity is increased in the case of a target material in its condensed phase, where non-local interactions induce a collective response of the medium as the projectile penetrates. The magnitude of the problem has demanded of the development of appropriate theoretical models whereby the relevant mechanisms of energy loss are judiciously introduced so that their relative participation in the stopping process allows understanding the experimentally observed behavior. In this spirit, experiment has played a paramount role to shed light on the dominant mechanisms of energy loss as a function of projectile/target type and energy. Moreover, the advent of more powerful and accurate experimental techniques has imposed
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.063
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further refinements to the theory in order to understand more subtle – yet important – details of the energy loss phenomenon such as the role of electron capture and loss mechanisms, band-gap effects in solid-state materials, physical phase-state effects and chemical binding effects to mention a few. Even though the theory of ion stopping has been developed with relative success in predicting range profiles and damage distribution for a wide class of materials leading to important applications in ion implantation techniques, still the above mentioned fundamental questions remain open for research. The purpose of this contribution is to present preliminary results on the behavior of the electronic stopping cross section (Se ) and total range of heavy ions penetrating a molecular target under high pressure. The motivation of this study has its root in recent work developed in our group [1–3] to treat consistently the effect of increasing target density (pressure) on chemical bond contributions to Se for protons. Indeed, if target density increases, one would expect that the electronic stopping cross section should be reduced according to its definition: Se ¼
1 dE ; n dx
ð1Þ
where n is the target number density and dE the dx electronic stopping power. It is not true, however, that the electronic stopping power for a target under pressure will remain the same as for the free target since this quantity is strongly dependent on the properties of its electronic structure. For a target under high pressures its electronic properties change strongly [2,3], hence a competing behavior between stopping power and target number density will yield a corresponding change in Eq. (1). In this work, the results obtained for proton stopping in dense molecular targets [2] are used to construct the corresponding electronic stopping cross section of heavy ions (SeHI ) using – as a first approximation – conventional effective charge scaling theory [4] whereby 2
SeHI ðvÞ ¼ ðcZHI Þ Sep ðvÞ;
ð2Þ
where c is the fractional effective charge term (see further below), ZHI the ion nuclear charge and Sep
the stopping cross section for protons moving with the same velocity ðvÞ as the heavy ion in a given medium. The total ion range is then estimated as a function of target pressure and compared with its corresponding value at normal pressure. Taking liquid water and methane under high pressure as sample media and Heþ (E 6 1:6 MeV) and Liþ (E 6 2:6 MeV) as projectiles, the associated pressure-dependent range-energy curves are constructed. It is found that the total range is effectively reduced as target pressure increases due the combined effect of target density increase and a corresponding decrease in SeHI . The work is organized as follows. In Section 2, a brief description of the theoretical assumptions and development for the treatment of molecular stopping under pressure is presented along with the results and a discussion of the main findings. In Section 3 the main conclusions of this study are drawn.
2. Theoretical considerations Consider a given molecular target in its condensed phase and subject to high pressure. For the purposes of our discussion, we shall assume here the simplest approach to a medium in its condensed phase, i.e. no account of phase diagramgoverned transitions and an equation of state (EOS) equally valid for the solid and the liquid phase. In this context, we consider the medium as formed by closely packed molecular entities each one of them spatially confined by its neighbors. Furthermore, we assume that the intermolecular potential is such that an infinitely high barrier potential appears at the boundary of the confinement volume. Accordingly, as pressure is increased the confinement volume for each molecule is reduced with consequent changes in the molecular electronic properties [1–3,5]. Since the inelastic energy loss mechanisms for an incoming projectile are inherently related to the projectile-target electronic structure, our goal is then to incorporate the changes in molecular electronic properties in the governing stopping power equation. Hence, a first step on call is to account for changes in the molecular properties as a function of pressure.
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
2.1. Model of molecular confinement According to our previous discussion, we shall consider a molecule in a dense medium as a system confined within an impenetrable spherical cavity whereby changes in pressure correspond to changes in cavity volume. In recent work we have discussed in detail the ab initio treatment of molecules confined by hard-walled spherical cages [3]. Here we give a brief summary of the main assumptions and results which are useful for the purposes of this work. Fig. 1 shows schematically a CH4 molecule within a spherical cavity of radius R. Clearly, the first change we may foresee for the confined molecule is a redistribution of its electronic density within the cavity. Indeed, this idea was first explored for the water molecule by Xu et al. [6] in their analysis of physical-phase state on proton stopping. These authors considered the electronic charge redistribution by renormalizing the freemolecule charge density within the confinement volume. However, the renormalized density is not consistent with the actual dynamical changes taking place for the molecule to attain its final stable conformation. Once the molecule is confined, the
R
r
r – Rk
electronic charge redistributes with consequences on the nuclear positions and molecular energy. This problem has been addressed [1–3,5] by treating the molecular ground state energy within an ab initio scheme using the floating spherical Gaussian orbital (FSGO) representation of localized molecular orbitals [7] modified such that for a confined molecule each electron pair in a given core, bond or lone-pair orbital is represented as [8] 2
Wk ðr Rk Þ ¼ Nk eak ðrRk Þ ð1 r2 =R2 Þ;
Fig. 1. Schematic diagram showing a methane molecule enclosed in a spherical cavity of radius R. The core and bond FSGO orbitals are shown as spheres of radius qk ¼ a1=2 , where k ak is the orbital parameter (see text). The orbital center positions Rk are also shown.
ð3Þ
where Nk is the normalizing constant, 1=2 ak R2 k
Nk ¼ ð4pÞ Z
e
R
2
2
e2ak r i0 ð4ak Rk rÞð1 r2 =R2 Þ r2 dr
1=2
0
ð4Þ and Rk the position of the center of orbital k relative to the origin and r the position of any point in space relative to the origin within the confinement space (see Fig. 1). ak in Eqs. (3) and (4) is an orbital parameter which, together with Rk , are determined after minimization of the total molecular energy for a given box radius. Note that the term in parenthesis in Eq. (3) is a cut-off term which guarantees that the wavefunction vanishes at the boundary. Hence, the energy minimization procedure described provides the equilibrium charge density and geometric molecular conformation for the cavity radius considered. Table 1 displays, as an example, the behavior of the ground state energy of the water and methane molecules as a function of confinement radius and pressure [2,3]. Pressure was obtained from an interpolated curve for the E versus R values as
ρk = α k – 1 / 2 Rk
413
P ¼
oE 1 oE ¼ : oV 4pR2 oR
ð5Þ
Fig. 2 shows the behavior of the orbital parameter ak and bond length as a function of cavity radius for the C–H and O–H bonds of methane and water, respectively. Since the FSGO orbital radius 1=2 is given as qk ¼ ak (from the argument in the exponential in Eq. (3)), from Fig. 2 we note that the orbitals become more compact as the cavity shrinks. Furthermore, for this isotropic compression, the bond length is reduced and – in the case
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Table 1 Ground state energy and associated pressure for the confined H2 O and CH4 molecules as a function of cavity radius calculated through the FSGO method adapted for confined systems [2,3] H2 O
CH4
R (a.u.)
Total energy (Hartrees)
0.85 1.0 1.25 1.5 1.75 2.0 2.5 3.0 4.0 5.0 6.0 10.0 1
Pressure (GPa) 5
6.8 · 10 2.9 · 105 8.3 · 104 2.8 · 104 1.1 · 104 4.4 · 103 940 254 28.8 4.9 1.1 0.015 104
1.983 )22.628 )43.390 )53.119 )58.097 )60.782 )63.114 )63.862 )64.200 )64.258 )64.275 )64.286 )64.287
orbital radius (a.u.)
α
-1/2 C-H
1.4 1.2
α
0.8 (a)
bond length (a.u.)
0.6
2
C-H
1.5
O-H
1 0.5 (b) 4
6 R (a.u.)
7.1 · 105 3.2 · 105 1.0 · 105 3.7 · 104 1.5 · 104 7.0 · 103 1.7 · 103 532 75.6 15.6 4.2 0.09 104
2.2. Proton stopping cross section in a dense medium
1
2
80.042 20.659 )4.114 )16.356 )23.075 )27.027 )31.011 )32.680 )33.727 )33.924 )33.967 )33.990 )33.992
-1/2
O-H
0
Pressure (GPa)
of water – the bond angle increases while for methane the bond angle remains the same [3]. Hence, molecular confinement not only affects the electron distribution but the whole molecular characteristics. This behavior will certainly have important consequences on the electronic stopping cross section as we discuss below.
1.8 1.6
Total energy (Hartrees)
8
10
Fig. 2. Behavior of (a) the orbital radius qk ¼ ak1=2 and (b) C– H and O–H bond lengths for water and methane as the confining radius is reduced [3].
We shall consider now the main elements leading to the calculation of the electronic stopping cross section for protons (Sep ) moving in a medium under high pressure. To this end, we resort to the orbital implementation of the kinetic theory of stopping [9–11]. An advantage of using this procedure is that we may evaluate the cores and bonds (CAB) contributions to Sep and directly take into account deviations from Bragg’s additivity rule [2,12,13]. In the kinetic theory of stopping [14] the electrons of the medium are not considered at rest but moving with a velocity distribution f ðv2 Þ, where v2 is the electron velocity relative to the laboratory system. In the frame of reference moving with a projectile of charge Z1 and non-relativistic laboratory velocity v1 , the relative velocity of the target scatterers is thus v0 ¼ v1 v2 . Hence, according to the kinetic theory of stopping the electronic stopping cross section, Se ðv1 Þ, may be written as [15]
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
Se ðv1 Þ ¼
Z
f ðv1 v0 Þ
v1 v0 S0 ðv0 Þ d3 v0 ; v 1 v0
ð6Þ
where S0 ðv0 Þ is the value for Se in the case v1 v2 (scatterers at rest). Making use of the orbital decomposition scheme proposed by Oddershede and Sabin [9–11]: X Se ðv1 Þ ¼ Se;k ðv1 Þ; ð7Þ k
where Se;k is the contribution from orbital ‘‘k’’, Se;k ðv1 Þ ¼
4pe4 Z12 Z2 Lk ðv1 Þ; mv21
ð8Þ
with m and e the electron mass and charge, respectively. Z2 is the total number of electrons of the molecular target and Lk the orbital stopping number, which in terms of the orbital mean excitation energy (Ik ) and its electron population (xk ) is given as [15] Z 1 2mv02 2 1 Lk ðv1 Þ ¼ pv1 Z2 xk ln dv0 Ik ak Z p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fk v21 þ v02 2v1 v0 cos h dðsin2 hÞ: 0
ð9Þ
quired in Eqs. (9) and (10) are directly dependent on the electron charge density, which may be constructed after Eqs. (3) and (4) once the molecular parameters have been optimized for specific confinement conditions as described in the previous subsection. The detailed calculations leading to explicit expressions for fk and Ik have been reported elsewhere [2]. For completeness and without digressing too much from the aims of this paper, here we give a summary of the main findings in [2]. Accordingly, fk and Ik have been shown to be X 2 2 fk ðkÞ ¼ ð2lþ1Þ Nk2 e2ak Rk p l Z R 2 ak r2 2 2 2 e il ð2ak Rk rÞjl ðkrÞð1 r =R Þr dr ; 0
ð11Þ where jl ðyÞ and il ðzÞ are the lth-order spherical Bessel function and modified spherical Bessel function p offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the first kind, respectively, and k ¼ ðm=hÞ v21 þ v02 2v1 v0 cos #, according to Eq. (9). Also Z R 2 2 2 e2ak r ð1 r2 =R2 Þ i0 ln Ik ¼ 2pNk2 e2ak Rk 0
1=2
Here ak ¼ ðIk =2mÞ and fk ðjv1 v0 jÞ ¼ fk ðjv2 jÞ is the velocity distribution of the scatterers in orbital‘‘k’’ relative to the projectile. We have used the Bethe term for S0 in Eq. (6). The orbital excitation energies, Ik , may be evaluated through the orbital local plasma approximation OLPA [16,17] as 1=2 Z 1 4pe2 h2 qðrÞ ln Ik ¼ q ðrÞ ln d3 r; ð10Þ xk C k m where qk ðrÞ is the local electronic density in configuration space and qðrÞ is the local total electron density obtained after taking the angular average of each orbital charge density [17]. vk corresponds to the scaling parameter originally proposed by Lindhard and Scharff [18] to account for polarization effects. We set vk ¼ 1 in our case. Note that C in Eq. (10) indicates integration within the confinement volume only. Clearly, the orbital velocity distribution fk as well as the orbital mean excitation energy Ik re-
415
ð4ak Rk rÞ ln½4pe2 h2 qðrÞ=m r2 dr;
ð12Þ
with the angularly averaged total local electron density qðrÞ given as qðrÞ ¼ 2ð1 r2 =R2 Þ2 X 2 2 2 Nj Nk eAjk Rjk ajk Kjk eajk r i0 ð2ajk Kjk ÞTjk ; j;k
ð13Þ where Tjk are the elements of the inverse overlap matrix [2,7] and the following definitions hold: ajk ¼ aj þ ak ; Rjk ¼ jRj Rk j;
Ajk ¼ aj ak =ajk ; Kjk ¼ jaj Rj þ ak Rk j=ajk :
ð14Þ
The structure of Eqs. (12) and (13) suggests an important dependence of the mean orbital excitation energy on the confinement radius (hence on pressure, according to Eq. (5)). The corresponding confinement effects on the total molecular mean ionization energy may be deemed using the orbital
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partition rule suggested by Oddershede and Sabin [10]: " cores bonds X 1 X ln Imolecule ¼ xi ln Iicore þ xj ln Ijbond N i j # lone-pairs X lone-pair þ xk ln Ik ; ð15Þ k
where N is the total number of electrons and xk the electron population of each core, bond and lone-pair orbital. Table 2 shows the orbital and total mean excitation energies for the water and methane molecules [2] as a function of pressure and the associated confinement radius. Clearly, as pressure increases, the mean excitation energy increases, as expected. This behavior brings as a consequence a corresponding reduction in the electronic stopping cross section as may be gathered after inspection of Eqs. (8) and (9). Fig. 3 shows, as an example, the predicted values for Sep as a function of energy for protons traversing water in its vapor phase (continuous curve) and for water under a 4.9 GPa pressure (dashed curve) [2]. The symbols in this figure correspond to experimental data for water in its gaseous phase [19] (solid triangles) and in its solid phase [20] (open circles). The well-known physical phase state effect is apparent in these experimental curves showing a lower value (10%) in Sep for the
condensed phase as compared to the gaseous phase. It is then reasonable to expect an increased lowering in the stopping curve for the high pressure case, as shown here. Fig. 4 shows the corresponding effect for protons incident on methane for various pressures as compared with the gaseous phase [2], where this effect becomes evident indicating also a small shift of the maximum of the sopping curve towards higher energies as pressure increases. 2.3. Scaling to heavy ion stopping Once Sep is obtained, the heavy ion stopping cross section (SeHI ) may be estimated using the effective charge scaling expression given by Eq. (2). In this work, the fractional effective charge (c) is calculated using the recommended expressions given by Ziegler et al. [4,21], based on the Brandt– Kitagawa theory [22]: 2 " 2 # 1 v0 K vF c ¼ q þ ð1 qÞ ln 1 þ 2:084 ; 2 a0 v 0 vF ð16Þ where q is the fractional charge state of the ion, v0 is the Bohr velocity, a0 the Bohr radius and vF the Fermi velocity of the medium, which is assumed as vF ¼ v0 in the case of organic compounds [21]. K in
Table 2 Core, bond, lone-pair and total mean excitation energies for the confined H2 O and CH4 molecules obtained from Eq. (12) for different confinement radii [2] H2 O
CH4
R (a.u.)
Icore
Ilone-pair
Ibond
Imolecule
Icore
Ibond
Imolecule
0.75 1.0 1.25 1.5 2.0 2.5 3.0 4.0 6.0 8.0 1
529.6 484.0 447.3 430.4 411.9 404.1 400.8 399.3 399.1 399.0 399.0
302.1 227.3 159.8 127.3 88.8 71.2 63.1 58.6 57.7 57.6 57.6
301.1 225.7 157.8 125.0 85.9 67.7 58.9 53.7 52.6 52.4 52.4
357.5 263.2 195.3 161.2 119.6 98.8 88.9 83.1 81.8 81.7 81.6
464.4 359.1 311.3 291.3 270.2 260.3 254.9 250.4 249.0 248.9 248.9
336.9 224.7 153.8 118.8 75.8 54.6 43.0 32.3 28.5 28.2 28.1
359.2 341.2 177.1 142.1 97.8 74.6 61.4 48.7 43.9 43.6 43.5
All energies are given in eV.
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
q ¼ 1 exp½ð0:803yr0:3 þ 1:3167yr0:6
30 H → H2O
þ 0:38157yr þ 0:008983yr2 Þ ;
25 Free 20
Se(10
-15
P=4.9 GPa
10 5 0
50
100
150 200 250 Energy (keV)
300
350
400
H+ → CH4 40 gas phase 15.7 GPa 75.6 GPa
with ai ¼ ð0:2865; 0:1266; 0:001429; 0:02402; 0:001135; 0:00475; i ¼ 0; . . . ; 5Þ. Fig. 5 shows the theoretical stopping curve of Heþ ions incident on water at normal pressure (continuous curve) and at P ¼ 4:9 GPa (long-dashed curve) as a function of projectile energy. Also shown are available experimental data for the gaseous phase (solid circles [23] and open triangles [24]) as well as for the solid (open circles [23]) and liquid (inverted solid triangles [24]) phases. The short-dashed curve corresponds to scaled values obtained from the gas-phase theoretical curve after
532 GPa 80
0
2
4
v/v0
6
8
10
Fig. 4. Predicted pressure effects on the proton electronic stopping cross section for a methane target under various high pressure conditions [2]. The experimental data correspond to the gas phase: solid diamonds [19], open triangles [32], open diamonds [33].
-15
0
ev cm2/ molecule)
10
Se(10
Se(10
-15
ev cm2/ molecule)
50
20
with yr ¼ vr being the relative velocity of the ion’s electrons to the Fermi velocity of the medium approximated in this case (for v > v0 ) as vr vð1 þ 1=5v20 =v2 Þ. For Heþ projectiles, Ziegler et al. [4] suggest the use of a corrected fractional effective charge (cHe ) obtained after a systematic analysis of experimental data, namely " # 5 X i 2 cHe ¼ 1 exp ai lnðEÞ ; ð19Þ i¼0
Fig. 3. Electronic stopping cross section versus energy curve for protons incident on water under a 4.9 GPa pressure (dashed curve). Continuous curve: theoretical prediction for the gasphase at normal pressure. Experimental data correspond to the gas (solid triangles [19]) and solid (open circles [20]) phases.
30
ð18Þ
2=3 ZHI vr =v0 ,
2
ev cm / molecule)
+
15
417
2=3
K ¼ 0:686a0
1=3
ð1 qÞ ZHI : 1 17 ð1 qÞ
ð17Þ
The fractional charge state of an ion is given by Ziegler et al. as [4]
+
H 2O
60 gas phase
50
"scaled to condensed phase"
40
4.9 GPa
30 20 10
Eq. (16) is the dynamical screening length defined in the Brandt–Kitagawa theory in terms of the ion nuclear charge ZHI as
He
70
0
1
2
3 4 Energy (MeV)
5
6
7
Fig. 5. Theoretical predictions for the stopping cross section of Heþ ions incident on water vapor at normal pressure (continuous curve) and for water under a 4.9 GPa pressure (long-dashed curve). The short-dashed curve corresponds to the gas-phase curve reduced in 10% to account for the condensed phase (see text). Also shown are experimental points for water: gas phase (solid circles, [23]; open triangles, [24]); solid phase (open circles, [23]); liquid phase (inverted solid triangles, [24]).
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S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
reducing its amplitude in 10% to represent the observed physical phase-state difference. This latter average curve will be used when comparing total ranges within the same physical phase-state in water. In this preliminary study we shall consider only the energy region 0 6 E ðMeVÞ 6 1:6 and 0 6 E ðMeVÞ 6 2:6 for for Heþ and Liþ projectiles, respectively. As a matter of fact, in the above energy regions the stopping curves allow for the simple representation: SeHe ¼ Kua ebu
ðu ¼ v=v0 Þ;
ð20Þ
where the parameters K, a and b are determined after a best fit procedure to each curve. Table 3 displays the values of these parameters for the projectile-target combinations and pressures considered here. Eq. (20) will be useful for the total path range calculations given further below. 2.4. Total ion path range calculation The total ion path length has been calculated using the well-known expression for total stopping power:
dE ¼ nðSe þ Sn Þ; dx
ð21Þ
with Sn the nuclear stopping cross section given by Ziegler et al. [4]. In terms of the reduced velocity u ¼ v=v0 , the above expression may be cast as u
du ¼ ½Se ðuÞ þ Sn ðuÞ =k; dx
ð22Þ
with k¼2
Mp ðIH =nÞ; me
ð23Þ
where Mp , me , IH and n are the projectile atomic mass, the electron mass, Rydberg energy and target number density, respectively. From Eq. (22) the total range becomes then Z 0 u du R ¼ k ; ð24Þ u0 Se ðuÞ þ Sn ðuÞ with u0 the initial ion velocity. Using the fitted theoretical Se values (Eq. (20)) for Heþ (E 6 1:6 MeV) and Liþ (E 6 2:6 MeV) projectiles incident on water and methane as well as the corresponding Sn estimates according to [4], the integral in Eq. (24) was numerically evaluated. Note, however, that the factor k in Eq. (24) involves a reciprocal dependence on the target molecule number density n, as indicated by Eq. (23). For water and methane under high pressure, n was estimated from the respective pressure-density equation of state (EOS) of the compressed solid [25–27] using the pressure values obtained with our confinement model. Accordingly, for water at 4.9 GPa, q ¼ 1:64 g cm3 ; n ¼ 5:68 1022 molec cm3 while for methane at 15.7 GPa: q ¼ 1:09 g cm3 ; n ¼ 4:24 1022 molec cm3 and at 75.6 GPa: q ¼ 1:66 g cm3 ; n ¼ 6:49 1022 molec cm3 . Figs. 6 and 7 show the calculated pressuredependence of the total-path range versus energy curves of Heþ and Liþ projectiles, respectively, in
Table 3 Values of the parameters required in Eq. (20) for the analytical evaluation of the calculated electronic stopping cross sections for helium and lithium projectiles incident on water and methane at high pressures (see text) H2 O (‘‘liquid’’)
CH4
Projectile
Pressure (GPa)
K
a
b
Heþ Liþ
104
85.592 102.288
1.597 1.724
0.667 0.629
K
a
b
Heþ Liþ
4.9
53.255 64.489
1.944 2.135
0.693 0.682
Heþ Liþ
15.7
127.152 153.121
1.606 1.731
0.743 0.708
Heþ Liþ
75.6
108.882 132.495
1.651 1.804
0.725 0.704
K is in units of 1015 eV cm2 /molecule.
S.A. Cruz / Nucl. Instr. and Meth. in Phys. Res. B 222 (2004) 411–420
maximum. This behavior might reflect the competing role between target density increase and reduction in stopping cross section due to pressure increase as suggested by Eq. (24). A more detailed analysis is necessary including other projectiletarget systems to assess a definite conclusion.
10 +
Projectile: He
Total Range (µm)
8
1 atm
H O ( "liquid" )
6
419
2
4.9 GPa
4 CH
3. Conclusions
15.7 GPa
4
2
75.6 GPa
0 0.2
0
0.4
0.6
0.8
1
1.2
1.4
1.6
Energy (Mev)
Fig. 6. Predicted total path range-energy curves for Heþ ions traversing water and methane under various pressure conditions.
10 Projectile: Li
+
1 atm
Total Range (µm)
8 H O ( "liquid" ) 2
6
4.9 GPa
4
CH
15.7GPa
4
2
75.6 GPa
0 0
0.5
1.5 1 Energy (MeV)
2
2.5
Target pressure effects on the electronic stopping cross section and ranges of heavy ions have been analyzed in terms of effective charge theory and a molecular confinement model to simulate the effect of pressure. The results of this work are preliminary and predict an important reduction of the stopping cross section and range of swift ions traversing a target under high pressure. While the differences are more noticeable for high ion energies and gigapascal pressures, it is deemed here that a possible scenario where these differences might be observed is the length of fission fragment tracks formed in materials under high pressure as compared to those formed in the same material at normal pressure [28]. Indeed, recent experiments on fission track length registration and annealing in apatite under high pressure seem to give evidence of this effect [29], which has been pointed out – with some debate – as crucial for thermochronology studies [30,31]. Further experimental and theoretical work is necessary to ascertain the general validity of the results presented here.
Fig. 7. Same as Fig. 6 for Liþ ions.
Acknowledgements water and methane. For Heþ on water, the ‘‘liquid’’ phase results at normal pressure were obtained through the scaled gas-phase Se values mentioned at the end of the previous section. From these figures we observe a systematic reduction in the range values as pressure increases. This reduction may be considerably large mainly for higher energies. Note, however, a slight crossover in the range values for low projectile energies (EðHeÞ 6 0:4 MeV; EðLiÞ 6 0:6 MeV) in the case of water in its condensed phase at P ¼ 1 atm and P ¼ 4:9 GPa. This effect may be related with the steep reduction in the corresponding Se versus E curves below their
I am deeply indebted to Lew Chadderton for bringing to my attention the problem of pressure effects on the range of swift ions. I also appreciate the illuminating and inspiring discussions with him during his attendance to the First International Radiation Effects Meeting in Mexico City which certainly enriched the contents of this contribution. References [1] S.A. Cruz, J. Soullard, E.G. Gamaly, Phys. Rev. A 60 (1999) 2207.
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