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Numerical evaluation of the electronic stopping power for heavy ions in solids
Nuclear Instruments and Methods in Physics Research B80/81 (1993) 16-19 North-Holland
i,
13
Beam Intsraetions with Materials S Atoms
Numerical evaluation of the electronic stopping power for heavy ions in solids Ynn-Nia^
Wang and Teng-Cad M,-:
The Key National Laboratory of Moterial Modifcanon by Beam Three, Dalian Unicerwy of 'Jrrhnology, Dalia, . 116023, Chma
The electronic stopping powers for heavy ions moving in solids are calculated within the framework of the linear-response dielectric theory and in the local-density approximation. The exchange-correlation int-,ction of electron, solid is taken into account with the local-field correlation dielectric function . An analytical expression of the solid-state electron density is presented . Theoretical results are compared with experimental data .
1. Introduction The investigation of the electronic stopping powers for heavy ions in solids is ~,cry important for the modification processes of material surfaces using ion beams . Due to the complexity of atomic structures, it is difficult with a single formula to calculate the stopping power of any projectile within. any target over a wide velocity range. There is a general trend relating the stopping power with velocity, and three diff=nt velocity regions correspond to different energy loss processes can be characterized. In the high-velocity region, the incident ion is stripped of all its electrons upon entering the target, ionization of individual target atoms is the main source of energy loss, and the Bothe-Bloch theory [1] is applicable to describe the process. In the low-velocity region, where both projectile and target atoms are practically neutral, the theories, based on statistical model [2,3], predict a stopping power proportional to velocity, which can be nicely pictured as a fictional process. In the intermediate-velocity region, the calculations are considerably more complicated. The ion is partially stripped and its slowing-down is influenced by the effect of electronic structures of target atoms . An empirical formula on the basis of a number of experimental data and the Brandt-Kitagawa (BK) theory [4] has been constructed by Ziegler, Biersack and Littmark (ZBL) [5]. In a previous work [6], with the local-field correcIlion (LFC) dielectric function we investigated the stopping powers for low-velocity ions in the valence-electron gases in solids . It has been shown that our results differ significantly from the pttdictions given by the random-phase approximation (RPA) dielectric theory. In the present work, we attempt to expand our previ-
ous work to calculate the stopping powers for any velocity heavy ions in solids. The contr'tutions of the inner-shell electrons of atoms in solids to the stopping power are included with the method of the local-density approximation . The effect of the electrons bound to the projectile is considered according to the B :{ model . 2. Dielectric formalism for electronic Qopping power Let us consider a heavy ion with velocity v moving in an homogeneous electron gas with d : nsity n . Within the framework of the linear-response .dielectric theory, the electronic stopping power can be written as [4] rk, S=2(,,1,2)-tlsp`(k) dk/k m dw o n xim[- 1/e(k, w)], where e(k, w) is the longitudinal dielectric function of the electron gas and p(k) is the Fourier transform of the charge distribution of the projectile. According to the BK model, p(k) is given by p(k) =Z,e[q+ (2kA)=]/[1 + 2kA)2 ], where Z, is the projectile's atomic number, e is the element charge, A = 0 .48nt,(1 -q )213Z, '/'/(1 -(1 -q)/7) is the screening length, and q is the ionization degree of the projectile calculated from the ZBL model [5]. In the following discussion the LFC dielectric function [7,8] e(k, w) = 1 -P(k, w) / [I +G(k)P(k, w)],
(2)
will be used, where G(k) is the LFC function to the random-phase approximation (RPA) dielectric function (in which G(k) = 0), which includes the correla-
0168-583X/93/$06.00 O 1993 - Elsevier Science Publishers B.V. All rights reserved
YW. Wang, T.-C Ma / Electronic stopping power for heavy ions in solids tion-exchange interaction of electrons, and P(k, w) is Lindhard's polarizability [9] . In the low-frequency (u =wlkV F 4 1) and longwavelength (z =k/2k F < 1) limit we obtain [6j
17
the collective and individual excitations of the electron gas. Using eq. (10) in eq . (1) we get Shoe
= (4Tre° /mv2)Zéttn,,L,
(11)
P(k, w) = -X2 [(I -z 2/3) +i7ru/21/z 2 ,
(3)
where the effective charge fraction stopping number are given by
G(k) =4YO z 2 ,
(4)
~2=q2+(1-q)((1-q)(xi-x2)
where k F =(3Tr 2n ) )'/ i is the Fermi wave number, vF=(k F a )v 0 is the Fermi velocity, X 2 = 1/(Trk F a ), and a = 0 .529 X 10 - s con and vo = 2.18 X 10' cm/s are the Bohr radius and the Bohr velocity, respectively . The parameter yo is connected to the correlation energy of the electron gas [7,8] . For low-velocity projectiles (v <" F), the stopping power can be approximately expressed as Ssoow =
( 4Tre °/mv 2 )Z~t n L,
(5)
whe,c 7_=- ;Z o is the effective charge, ~ i: the effective charge traction given by ~ -= [(q-tt )-+ (1 -q)-1(1/t0)l1(t) +2(1 - q)(q-tt )K(t, to)l1(t) ] /(1 - tt o )- , (6) in terms of the functions 1(z) = In(1 + 1/z) - 1/(1 +z),
(7)
K(t, t(,) = [In( 1 +t ) -tt(, In( I + 1/t)]/(1 -tt ) (8) and variables t=X 2/(1-ßX2 /3) and t =(2k F A) 2, ,6 = 1 + 12y, In eq. (5) L is the stopping number -,6X 2 13) 1(t ) . (9) L = 2 (t'IVF)3(1 -2 When 13 = 0 is assumed in eq. (5), the expressions of and L can be reduced to BK results [41 based on the RPA theory. For ordinary metals and semiconductors, however, the values of R range from 4 to 5 (6]. It has been shown that the effect of the correlation and exchange interaction of electrons acts to enhance the stopping power of slow projectiles [6,8] . At high velocities v > VF, the LFC effect in the dielectric function can be neglected and an approximate expression of the dielectric function can be neglected and an approximate expression of the dielectric function is given by [9]
Zeff
e(k,w)=1+top/ [(hk 2 /2m) 2 -(w+iS) 2 1,
(10)
where wp =(4Tre'n o/m) o /2 is the plasma frequency and S is an infinitesimally small positive quantity . This approximation corresponds to the case w/k ~" VF, but otherwise arbitrary k, and describes in very simple way
= Z~tt/Z, and the
+(1 +q) In[(1 +x 2 )/(1 +x,) ] }/(2L),
(12)
and L = In(2mv 2/hw p ) .
(13) In eq. (12) x, and x 2 are defined by x, =(AWP /v) 2 and x2 = (2Av/aov )2. In order to simplify the computational course presented in the following section, it is valuable to obtain an analytical expression of the stopping power in any velocity region . Here, we present a fitted function to the numerical results based on eq. (1) as follows: S
~S,1 _1(1 +
0 .8v/vF)
v 5 l .5V F v > 1.5"F
Shieh
(14)
This fitted function will be used in the following section . 3 . The local-density approximation In fact, the contributions to the stopping power are from both of the outer-shell electrons (i .e. valence electrons) and the inner-shell electrons of the target atoms, specially for high-velocity projectiles. For including the contributions coming from the inner-shell electrons the local-density approximation has been applied to calculate the stopping power [10]. in which the local electron density n(r) is used instead of the homogeneous electron-gas density n o . In this approximation the averaged stopping power can be written as S=N fR,,S[n(r)]4Trr 2 dr, 0
(15)
where N is the atomic density of the target, R 0 = [3/(4arN)]ors is the atomic radius, and the stopping power S[n(r)] is given by eq . (14). It is necessary to know the solid-state electron density for calculating the stopping power (eq . (15)) . The electron density in the solid differs from that of a free atom due to the overlap of the electronic wave function . A simple model of the solid-state electron density n(r) has been presented by Kaneko [I l] n(r) -
no {nA(r)
r>_Rc r
(16) fa. BASIC INTERACTIONS (a)
18
Y.-N. Wang. T.-C. Ma / Electronic stopping powerforheavy ions in solids
Mg in Ta
6
6
3
0
5
v/v Fig. 1 . The electronic stopping power for Mg ions in Ag target. The solid line corresponds to our calculations, the dashed line is the result of ZBL empiricA formula, and dots represent experimental data lief. [14]) .
where n,,(r) is the free-atom electron density, the valence-electron density no can be derived from experimental measurement of the plasma frequency [5,12], and Rc is fixed by the following continuity condition: tr~(Rc) = no .
(17)
With the atomic independent model (IPM) given by Green et al . [13], a parameterized expression of the free-atom electron density will be used in present work . The expression of nA(r) is found by n A( r
__ )
Z, H 4Trd a1)r
exp(17) H
I + 2(H - 1) I +H8
(
IS
)
where ZZ is the atomic number of the target, ,1 = r/'(dao), S=exp(q)-!, H=d(ZZ-1)o^, and d is a parameter determined by the Hartree-Fock-Slater screening functions and eigenvalues.
v/v Fig. 2. The electronic stopping power for Mg ions ir. Ta target . Same notation as in fig. l .
mx 8
x
v/'o Fig. 3. The electronic stopping power for C ions in Au target. The solid line corresponds to our calculations, (0) from ref. [15], (x) from ref. [161 and (,L) from ref. [17i represent experimental data .
4. Comparison of theoretical results with experimental data Using the inverted Doppler-shift-attenuation analysis method Arstila et al. [14] measured the electronic stopping power for Mg ions in 17 elemental solids in the velocity region v = 0.2vo-5vo. Figs. 1 and 2 show the velocity dependence of the stopping power for Mg ions in Ag and Ta targets. The solid and dashed lines correspond to our calculations predicted by eq . (15) and the results of the ZBL empirical formula, respectively, and dots represent the experimental data. At
low velocities v < 2v o, our results agree rathe , well with the data, but the ZBL predictions are obviously lower than the data . At high velocities 2vo < v < 5vo, ZBL results are closer to experimental data than our results.
mx É u 9 v
x
A] in Cu
v/v~ Fig. 4. The electronic stopping power for AI ions in Cu target. Same notation as in fig . 3,
Y.-.N. Wang, T.-C. Ma /Electronicstopping powerforheavy ions in solids In figs. 3 and 4 the comparison of our calculations with the experimental data [15-171 is made for C ions in An target and AI ions in Cu target . We can see from
both figures that our calculations are very close to the data for a wide variety of projectile-velocity regions u =vo12vo, especially for C ions in Au target . Simi-
larly to our work, Abdesselam et al . [171 have calculated the stopping power for C and Al ions in solids using the RPA dielectric function and the model of solid-state electron density given by Gertner et al . [181 . Their work reproduces the experimental data above the maximum of the stopping curve rather well . At low energy, however, a discrepancy is observed and its magnitude varies with target. These deviations might be attributable to inaccuracies of the RPA dielectric function and the density model of Gertner et al .
In conclusion, we have calculated the electronic stopping power for heavy ions in solids with the LFC dielectric function and the local-density approximation . It has been shown that our results are in good agreement with the available experimental data.
Acknowledgement This work has been supported by the Foundation of the "863-Advanced Technology" of China.
19
References [11 [2] [3] [4] [5)
See e.g., S.P. Ahlen, Rev. Mod. Phys . 52 (1980) 121. J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. O.B . Firsov, Sov . Phys . JETP 36 (1959) 1076. W. Brandt and M. Kitagawa, Phys . Rev. B25 (1982) 5631 . J.F. Ziegler, J.B . Biersack and U. Littmark, The Stopping and Ranges of Ions in Matter, vol . 1 (Pergamon, New York, 1985). [61 Y.N . Wang and T.C. Ma, Phys . Rev. A44 (1991) 1678 [7] K. Utsumi and S. Ichimaru, Phys. Rev. A2fi (1982) 603. [8] Y.N . Wang and T.C. Ma, Nucl . Instr. and Meth . B51 (1990) 216. [9] J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd . 28(8)(1954) [101 J. Lindhard and A. Winther, K. Dan. Vidensk . Selsk. Mat. Fys. 34(4)(1964). [I1] T. Kaneko, Phys. Rev. A30 (1984) 1^,14 . [121 D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964). [131 A.E.S . Green, D.L. Sellin and A.S . Zachor, Phys . Rev. 184 (1969) 1. [141 K. Arstila, J. Keinonen and P. Tikkanen, Phys. Rev. B41 (1990) 6117. [151 D.I . Porat and K. Ramavataram, Proc . Phys . Soc. 77 (1961) 97 . [161 D.C . Santry and R.D. Werner, Nucl. Insu. and Meth . B53 (1991) 7. [171 M. Abdesselam et al., Nucl . Instr. and Meth. B61 (1991) 385. [18] 1 . Gertner, M. Meron and B. Rosner, Phys . Rev. A18 (1978) 2022.
Ia . BASICINTERACTIONS (a)
i,
13
Beam Intsraetions with Materials S Atoms
Numerical evaluation of the electronic stopping power for heavy ions in solids Ynn-Nia^
Wang and Teng-Cad M,-:
The Key National Laboratory of Moterial Modifcanon by Beam Three, Dalian Unicerwy of 'Jrrhnology, Dalia, . 116023, Chma
The electronic stopping powers for heavy ions moving in solids are calculated within the framework of the linear-response dielectric theory and in the local-density approximation. The exchange-correlation int-,ction of electron, solid is taken into account with the local-field correlation dielectric function . An analytical expression of the solid-state electron density is presented . Theoretical results are compared with experimental data .
1. Introduction The investigation of the electronic stopping powers for heavy ions in solids is ~,cry important for the modification processes of material surfaces using ion beams . Due to the complexity of atomic structures, it is difficult with a single formula to calculate the stopping power of any projectile within. any target over a wide velocity range. There is a general trend relating the stopping power with velocity, and three diff=nt velocity regions correspond to different energy loss processes can be characterized. In the high-velocity region, the incident ion is stripped of all its electrons upon entering the target, ionization of individual target atoms is the main source of energy loss, and the Bothe-Bloch theory [1] is applicable to describe the process. In the low-velocity region, where both projectile and target atoms are practically neutral, the theories, based on statistical model [2,3], predict a stopping power proportional to velocity, which can be nicely pictured as a fictional process. In the intermediate-velocity region, the calculations are considerably more complicated. The ion is partially stripped and its slowing-down is influenced by the effect of electronic structures of target atoms . An empirical formula on the basis of a number of experimental data and the Brandt-Kitagawa (BK) theory [4] has been constructed by Ziegler, Biersack and Littmark (ZBL) [5]. In a previous work [6], with the local-field correcIlion (LFC) dielectric function we investigated the stopping powers for low-velocity ions in the valence-electron gases in solids . It has been shown that our results differ significantly from the pttdictions given by the random-phase approximation (RPA) dielectric theory. In the present work, we attempt to expand our previ-
ous work to calculate the stopping powers for any velocity heavy ions in solids. The contr'tutions of the inner-shell electrons of atoms in solids to the stopping power are included with the method of the local-density approximation . The effect of the electrons bound to the projectile is considered according to the B :{ model . 2. Dielectric formalism for electronic Qopping power Let us consider a heavy ion with velocity v moving in an homogeneous electron gas with d : nsity n . Within the framework of the linear-response .dielectric theory, the electronic stopping power can be written as [4] rk, S=2(,,1,2)-tlsp`(k) dk/k m dw o n xim[- 1/e(k, w)], where e(k, w) is the longitudinal dielectric function of the electron gas and p(k) is the Fourier transform of the charge distribution of the projectile. According to the BK model, p(k) is given by p(k) =Z,e[q+ (2kA)=]/[1 + 2kA)2 ], where Z, is the projectile's atomic number, e is the element charge, A = 0 .48nt,(1 -q )213Z, '/'/(1 -(1 -q)/7) is the screening length, and q is the ionization degree of the projectile calculated from the ZBL model [5]. In the following discussion the LFC dielectric function [7,8] e(k, w) = 1 -P(k, w) / [I +G(k)P(k, w)],
(2)
will be used, where G(k) is the LFC function to the random-phase approximation (RPA) dielectric function (in which G(k) = 0), which includes the correla-
0168-583X/93/$06.00 O 1993 - Elsevier Science Publishers B.V. All rights reserved
YW. Wang, T.-C Ma / Electronic stopping power for heavy ions in solids tion-exchange interaction of electrons, and P(k, w) is Lindhard's polarizability [9] . In the low-frequency (u =wlkV F 4 1) and longwavelength (z =k/2k F < 1) limit we obtain [6j
17
the collective and individual excitations of the electron gas. Using eq. (10) in eq . (1) we get Shoe
= (4Tre° /mv2)Zéttn,,L,
(11)
P(k, w) = -X2 [(I -z 2/3) +i7ru/21/z 2 ,
(3)
where the effective charge fraction stopping number are given by
G(k) =4YO z 2 ,
(4)
~2=q2+(1-q)((1-q)(xi-x2)
where k F =(3Tr 2n ) )'/ i is the Fermi wave number, vF=(k F a )v 0 is the Fermi velocity, X 2 = 1/(Trk F a ), and a = 0 .529 X 10 - s con and vo = 2.18 X 10' cm/s are the Bohr radius and the Bohr velocity, respectively . The parameter yo is connected to the correlation energy of the electron gas [7,8] . For low-velocity projectiles (v <" F), the stopping power can be approximately expressed as Ssoow =
( 4Tre °/mv 2 )Z~t n L,
(5)
whe,c 7_=- ;Z o is the effective charge, ~ i: the effective charge traction given by ~ -= [(q-tt )-+ (1 -q)-1(1/t0)l1(t) +2(1 - q)(q-tt )K(t, to)l1(t) ] /(1 - tt o )- , (6) in terms of the functions 1(z) = In(1 + 1/z) - 1/(1 +z),
(7)
K(t, t(,) = [In( 1 +t ) -tt(, In( I + 1/t)]/(1 -tt ) (8) and variables t=X 2/(1-ßX2 /3) and t =(2k F A) 2, ,6 = 1 + 12y, In eq. (5) L is the stopping number -,6X 2 13) 1(t ) . (9) L = 2 (t'IVF)3(1 -2 When 13 = 0 is assumed in eq. (5), the expressions of and L can be reduced to BK results [41 based on the RPA theory. For ordinary metals and semiconductors, however, the values of R range from 4 to 5 (6]. It has been shown that the effect of the correlation and exchange interaction of electrons acts to enhance the stopping power of slow projectiles [6,8] . At high velocities v > VF, the LFC effect in the dielectric function can be neglected and an approximate expression of the dielectric function can be neglected and an approximate expression of the dielectric function is given by [9]
Zeff
e(k,w)=1+top/ [(hk 2 /2m) 2 -(w+iS) 2 1,
(10)
where wp =(4Tre'n o/m) o /2 is the plasma frequency and S is an infinitesimally small positive quantity . This approximation corresponds to the case w/k ~" VF, but otherwise arbitrary k, and describes in very simple way
= Z~tt/Z, and the
+(1 +q) In[(1 +x 2 )/(1 +x,) ] }/(2L),
(12)
and L = In(2mv 2/hw p ) .
(13) In eq. (12) x, and x 2 are defined by x, =(AWP /v) 2 and x2 = (2Av/aov )2. In order to simplify the computational course presented in the following section, it is valuable to obtain an analytical expression of the stopping power in any velocity region . Here, we present a fitted function to the numerical results based on eq. (1) as follows: S
~S,1 _1(1 +
0 .8v/vF)
v 5 l .5V F v > 1.5"F
Shieh
(14)
This fitted function will be used in the following section . 3 . The local-density approximation In fact, the contributions to the stopping power are from both of the outer-shell electrons (i .e. valence electrons) and the inner-shell electrons of the target atoms, specially for high-velocity projectiles. For including the contributions coming from the inner-shell electrons the local-density approximation has been applied to calculate the stopping power [10]. in which the local electron density n(r) is used instead of the homogeneous electron-gas density n o . In this approximation the averaged stopping power can be written as S=N fR,,S[n(r)]4Trr 2 dr, 0
(15)
where N is the atomic density of the target, R 0 = [3/(4arN)]ors is the atomic radius, and the stopping power S[n(r)] is given by eq . (14). It is necessary to know the solid-state electron density for calculating the stopping power (eq . (15)) . The electron density in the solid differs from that of a free atom due to the overlap of the electronic wave function . A simple model of the solid-state electron density n(r) has been presented by Kaneko [I l] n(r) -
no {nA(r)
r>_Rc r
(16) fa. BASIC INTERACTIONS (a)
18
Y.-N. Wang. T.-C. Ma / Electronic stopping powerforheavy ions in solids
Mg in Ta
6
6
3
0
5
v/v Fig. 1 . The electronic stopping power for Mg ions in Ag target. The solid line corresponds to our calculations, the dashed line is the result of ZBL empiricA formula, and dots represent experimental data lief. [14]) .
where n,,(r) is the free-atom electron density, the valence-electron density no can be derived from experimental measurement of the plasma frequency [5,12], and Rc is fixed by the following continuity condition: tr~(Rc) = no .
(17)
With the atomic independent model (IPM) given by Green et al . [13], a parameterized expression of the free-atom electron density will be used in present work . The expression of nA(r) is found by n A( r
__ )
Z, H 4Trd a1)r
exp(17) H
I + 2(H - 1) I +H8
(
IS
)
where ZZ is the atomic number of the target, ,1 = r/'(dao), S=exp(q)-!, H=d(ZZ-1)o^, and d is a parameter determined by the Hartree-Fock-Slater screening functions and eigenvalues.
v/v Fig. 2. The electronic stopping power for Mg ions ir. Ta target . Same notation as in fig. l .
mx 8
x
v/'o Fig. 3. The electronic stopping power for C ions in Au target. The solid line corresponds to our calculations, (0) from ref. [15], (x) from ref. [161 and (,L) from ref. [17i represent experimental data .
4. Comparison of theoretical results with experimental data Using the inverted Doppler-shift-attenuation analysis method Arstila et al. [14] measured the electronic stopping power for Mg ions in 17 elemental solids in the velocity region v = 0.2vo-5vo. Figs. 1 and 2 show the velocity dependence of the stopping power for Mg ions in Ag and Ta targets. The solid and dashed lines correspond to our calculations predicted by eq . (15) and the results of the ZBL empirical formula, respectively, and dots represent the experimental data. At
low velocities v < 2v o, our results agree rathe , well with the data, but the ZBL predictions are obviously lower than the data . At high velocities 2vo < v < 5vo, ZBL results are closer to experimental data than our results.
mx É u 9 v
x
A] in Cu
v/v~ Fig. 4. The electronic stopping power for AI ions in Cu target. Same notation as in fig . 3,
Y.-.N. Wang, T.-C. Ma /Electronicstopping powerforheavy ions in solids In figs. 3 and 4 the comparison of our calculations with the experimental data [15-171 is made for C ions in An target and AI ions in Cu target . We can see from
both figures that our calculations are very close to the data for a wide variety of projectile-velocity regions u =vo12vo, especially for C ions in Au target . Simi-
larly to our work, Abdesselam et al . [171 have calculated the stopping power for C and Al ions in solids using the RPA dielectric function and the model of solid-state electron density given by Gertner et al . [181 . Their work reproduces the experimental data above the maximum of the stopping curve rather well . At low energy, however, a discrepancy is observed and its magnitude varies with target. These deviations might be attributable to inaccuracies of the RPA dielectric function and the density model of Gertner et al .
In conclusion, we have calculated the electronic stopping power for heavy ions in solids with the LFC dielectric function and the local-density approximation . It has been shown that our results are in good agreement with the available experimental data.
Acknowledgement This work has been supported by the Foundation of the "863-Advanced Technology" of China.
19
References [11 [2] [3] [4] [5)
See e.g., S.P. Ahlen, Rev. Mod. Phys . 52 (1980) 121. J. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128. O.B . Firsov, Sov . Phys . JETP 36 (1959) 1076. W. Brandt and M. Kitagawa, Phys . Rev. B25 (1982) 5631 . J.F. Ziegler, J.B . Biersack and U. Littmark, The Stopping and Ranges of Ions in Matter, vol . 1 (Pergamon, New York, 1985). [61 Y.N . Wang and T.C. Ma, Phys . Rev. A44 (1991) 1678 [7] K. Utsumi and S. Ichimaru, Phys. Rev. A2fi (1982) 603. [8] Y.N . Wang and T.C. Ma, Nucl . Instr. and Meth . B51 (1990) 216. [9] J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd . 28(8)(1954) [101 J. Lindhard and A. Winther, K. Dan. Vidensk . Selsk. Mat. Fys. 34(4)(1964). [I1] T. Kaneko, Phys. Rev. A30 (1984) 1^,14 . [121 D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964). [131 A.E.S . Green, D.L. Sellin and A.S . Zachor, Phys . Rev. 184 (1969) 1. [141 K. Arstila, J. Keinonen and P. Tikkanen, Phys. Rev. B41 (1990) 6117. [151 D.I . Porat and K. Ramavataram, Proc . Phys . Soc. 77 (1961) 97 . [161 D.C . Santry and R.D. Werner, Nucl. Insu. and Meth . B53 (1991) 7. [171 M. Abdesselam et al., Nucl . Instr. and Meth. B61 (1991) 385. [18] 1 . Gertner, M. Meron and B. Rosner, Phys . Rev. A18 (1978) 2022.
Ia . BASICINTERACTIONS (a)