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Local-field corrections in the calculation of electronic stopping power for protons
Nuclear Instruments and Methods in Physics Research B12 (1985) 34-37 North-Holland, Amsterdam
34
LOCAL-FIELD CORRECTIONS FOR PROTONS
IN THE CALCULATION
OF ELECTRONIC
STOPPING
POWER
I. NAGY, J. LASZLb and J. GIBER Department
of Atomic Physics, Technical University Budapest, Budafoki w 8., H- 111 I Budapest, Hungary
Received 19 October 1984 and in revised form 14 December 1984
The energy loss of slow protons in a degenerate electron gas is calculated on the basis of the dielectric theory. The influence of various longitudinal dielectric functions is examined. Comparison with the results of the kinetic theory is made. The effect of the local-field correction on the straggling parameter is also dibsed.
1. Energy loss
a = (4/9n)1’3
The change of the kinetic energy (W) per unit path length of a proton (regarded as a point charge) moving in a dielectric medium and having velocity u at a certain point of the trajectory is given by the well known formula u-‘(dW/dt)=u-’
euE(r,
t)l_,,,,
(1)
where E(r, t) is the space and time dependent electric field strength generated by the moving charged particle. Formula (1) is equivalent to the one of Flores and Garcia Moliner [l] applying the induced potential u-‘(dW/dt)
= --u-l
e[a+(r,
t)/i3 t] (,=vt.
Imfdk
vk/e(nk,
k) k*,
(2)
where the quantity S introduced denotes the stopping power usually used in particle-solid interactions. E is the dielectric function of the medium, k is the wave vector. The dielectric function of a degenerate electron gas at absolute zero temperature is defined [2] as e( 0, k) = I + Q&l-
Q&h
(3)
where o = uk, Q, is Lindhard’s complex polarizability [6], G is the so called local-field correction. Both Qa and G depend on w and k. The dependence of G on o can be neglected if u QCur ( ur is the Fermi velocity) Introducing
the usual notations
x2 = e*/rhu,
= arJa,
0168-583X/85/$03.30 8 Elsevier Science Publishers (North-Holland Physics Publishing Division)
-KC(r,)u=
density
=L’dz
remains
the first Bohr
--ylu
can be obtained, where electron mass. Thus C(r,)
and a,
z’/{
(4) K = 4e4m%/3ah3,
z* -x2
to be determined,
A(z)
[G(z)
m,
- II}’
is the
(5)
where
ji(z)=1/2+(1-z2)(4z)-11n((z+l)/(z-1)]. (6) 2. Analysis of C(r,) 2.1. The case of G=O Several approximate forms are known in the literature. If the so called small-k approach for E is used in first order (i.e. ji = l), then C(r,)=
[ln(l
+1/X2)-(1
+x2)-*l/2
(7)
results. This shape, shown as curve 1 in fig. 1, is known [3,4] as the modification of the early Fermi-Teller formula C(r,) - ln(l/x*)[S]. If ji is expanded up to second order (j, = 1 - z*/3), the result is still an analytical formula C(r,)=9(3
-X2)-*{ln[(2 - (3 - x2)/(3
x*-t + 2x2)}/2
3)/3
x2] (8)
(curve 2 in fig. 1). Using the complete function ji the Lmdhard-Winther result [6] is obtained after numerical integration (curve 3 in fig. 1). It is obvious from the theory of the quantum-mechanical many-body problem,
PI. z = k/2k,,
S,=
n is the electron
(1’)
If the medium is homogeneous and infinite, and if the change of the field is slight enough (i.e. if Maxwell’s laws of linear electrodynamics can be used), then the following equation can be deduced from (1) after the Fourier transformation of the space variables S, = e* (2a*u)-1
where radius
and rsa, = (3/4~n)l’~,
B.V.
I. Nagv et al. / Local-field corrections
35
function of Silin [13] is applied to arrive at
C(rJ
‘I~ cl&J
C( rs) = (1 + Fi/3)(1
1.5 :
X [ln(l+
+ F:)-*
l/x:)-{1
+xI)-J/2,
(9)
which is formally very similar to (7). Here x;=xz(l+F;)-l
(l+F;/3)-‘.
According to Pines and Nozieres [7], Fz and F; can be determined from the effective mass formula mz/m,
= 1 -I- F[/3
and from the compressibility x0/x = (1+ F;),‘(l
0
i 0
. 0.5
1
Z2a0.166rS
Fig. 1. Various approximations for C( r*) functions (eq. (5)). Curve 1: f, = 1, G = 0. Curve 2: fi = 1- z*/3, G = 0. Curve 3: Lindhard-Winther with compiete fi, G = 0. Curve 4: complete fr, G(z) after Ikvreese et al. Curve 5: complete ft, G(z, r*) afte Lantto et al. Curve 6: results of density-functional formalism after Echenique et al. Circles: results of Landau theory of Fermi liquids for ‘; = 1,2, 3and4.
2.2. ThecaseofG+O.
2.3. Application of binary collision theory From the theory in which the proton-electron interaction is considered as a pure binary collision (Finneman) 1161,Sigmund [17]) m, eti(kr)u
(10)
can be obtained, where n is the gas density, transport cross-section and v SE ur. Applying n = k;/3n2, and
The exchange and correlation contributions are taken into account by the local-field correction function. The calculations of Devreese et al. [2] and Tripathy and Mandal [&] considering only the exchange cont~bution result in a universal G(z) function (i.e. not depending on rs). For small z values G(z) - z’, while at z z 1 (i.e. k = 2k,) G has a sharp maximum (G,, P 2). Using (5), this correction leads to a C(r,) curve (4 in fig. 1) with a minimum (contrary to the result of Sayasov [9]). This is unexpected from the physical point of view, since an electron gas of lower density should not slow down a proton more efficiently. According to theories that consider also the correlation [lo-121, G(z) has no such sharp maximum and G_ zs 1, if z = 1 (G(z) - z2 is still valid for small z values). Also, G shows a slight dependence on r,. Using the data of Lantto et al. [lo] for G(z, rS), our resulting C(r,) can also be seen fig. 1 (curve 5). At last a numerical calculation is shown, which is based on the Landau theory of Fermi liquids. The E
-t-F;),
where x0 and x are the ~mpressib~ties of the free and the correlated electron gas, respectively. For the theoretical determination of these quantities, the exact value of the correlation energy has to be known [14,15]. Using the data of Pines and Notieres [7] our corresponding I values from eq. (9) (for r, = 1,2, 3 and 4) are shown in fig. 1 by circles . These points fully agree with the function values calculated with the help of G(z, rS) after Lantto et al. [lo].
Sn = --n+ that the Lindhard function is only applicable if r, -=K1 E71.
ratio
vF = e2/nhX2,
au(kF)=*
f
(‘Itl)
a,, the
kF = rng&h
sin2(6r--SI+,)
k6 I-O in eq. (10) we obtain sa=
-KC’(r,)v
= -yr 0,
where C’(r,)=(~x~)-~~
(11) (I+l)
sin2 (al-a,,,)
with
8, being the phaseshifts’:fO the electron scattering on a screened proton and K is the same constant as in eq. (4). For this calculation knowledge of the scattering potential is necessary. Using the static (I(W+ 0, k) with fi = 1, the usual Yukawa (Debye) type screened potential can be obtained (in atomic units V(r) = 2 exp (-kor)/r, where k. = 2/(a’~~)~/*). In the first Born approximation with this potential one would obtain exactly formula (7), since ,go(f+
1) sin2@,-&+,)
= ( nX2)2[ ln(1 + l/x’)
- (1+ x2)-l]
/2.
STOPPING POWER WORKSHOP
I. Nagy et al. / Local-field corrections
36
This result is not surprising, as in the region where the Lindhard function can be used (r, -K 1) the Fermi velocity ur is high (i.e. ur 3* e2/h), and the Born approximation is good for high energies. The screened potential of the proton has been calculated self-consistently by Echenique et al. [4] within the density-functional formalism. Afterwards the shift factors are determined numerically [HI. This leads to curve 6 in fig. 1. It should be noted that in this case C’(r,) decreases more strongly with increasing r,. This is to be expected, since the induced screened potential is the result of a non-linear calculation exceeding the linear response theory. So as r, increases, the energy loss of the proton decreases more rapidly (within this theory) than is predicted by the linear theory. This is due to the occurrence of bound states of atomic character which tend to screen interactions with the electron gas.
deviation is of the order of a few percent. The various approximations to the Lindhard function (eq. (6)) which gave very similar curves for C( rs) now give considerably different functions C,(r+): curve 3 (corresponding to fr =I 1 - z2/3) or curve 4 (where fr = 1). Jn these latter cases C2( rs) can be expressed analytically.
The measurable energy change of a proton passing through a metal foil of thickness Ax is - AW = W, W( AX), where W, is the primary energy, W(Ax) is the transmitted particle energy. If formula (2) or (11) is considered a simple Newton equation of motion and the nuclear stopping is neglected, then mP t=
-yv
(13)
results. Thus, the kinetic energy after penetration depth Ax is
3. Straggling As could be seen above, the introduction of G(z) f 0 causes an essential change in the C(r,) curves in comparison to the result of the Lindhard-Winther theory. This is not so for the straggling parameter. According to Sigmund [17] the straggling parameter (T) in the low velocity range is
W(Ax)
(12) where the formula for C,(r,) is similar to eq. (5) but z4 must be substituted for z3. The calculation of C,( r,) with the complete f*(z) function (eq. (6)) results in curve 1 in fig. 2. Considering G(z, rs) after Lantto et al. [lo], we obtain curve 2 in fig. 2. It can be seen that the
C,(rs) i
uo)‘,
(14)
where va is the initial velocity. If J%$ is given in keV, and Ax in A, then the following simple formula can be obtained from (14) W(Ax)=
T= 4ae4$(u/u,)2C,(r,),
= W,(l - yAx/m,
into
Wo[l -2.2X1O-3
C(rs)AxfW~‘z]2
(14)
In the measurements of Blume et al. [19] the gold target was bombarded by protons in channeling directions, In this case r, = 1.49 (i.e. x2 = 0.247) [20]. Our results for C(r,) from eq. (9) (or from (5) using the local-field correction proposed by Lantto et al. [lo]) is C( r.) = 0.7. (According to the Lindhard theory this value would be 0.46.) Table 1 contains the calculated and experimental W(Ax) values for a pair of primary energies and target thicknesses. Our theoretical results show very good agreement with the measured data within experimental error. Some more physical effects should be taken into account for developing the theory. Such phenomena are the surface effects at the place of entrance and escape of the projectile (e.g. for metals, the surface plasmon generation) [21,22], the possible inhomogeneity of the elecTable 1 Experimental and theoretical transmitted particle energy valttes for gold target
0
4 0
e
0.5
1
?&0.166 rs
Fig. 2. Various approximations for C,( );) functions. Curve 1: L~dh~d-umber with complete _f,, G = 0. Curve 2: complete II, G(r, rS) after Lantto et al. Curve3:f,=l-r2/3, GiO. Curve4:f,=l,
G%O.
Target thickness Ax
[A]
620 1200
Primary energy
Transmitted particle energy -. W(Ax) [keVl
JGEkeVlExp.[19]
Theory (present)
Theory darned)
6 18
2.25 5.75
3.32 9.17
- 2.5 : 6
I. Nags et al. / Local-field
tron gas and moreoverT in the case of LDA (locai-density a~~ro~mation~ f23], the gradient of the electron density close to the lattice atoms. If the bombardment is done avoiding cbmeling directions, an additive speed change of the projectile takes place when passing ‘close to a target atom due to the Coulomb interaction [24]. In relation to the LDA theory we must add that the determination of the electron density induced by a given proton-solid atom interaction can be made only by the solution of the time dependent SchrBdinger equation. In the case of slow velocities, this problem leads to the quasistationary problem for a quasimolecub. The usual atomic form of the electron density of the target atom is valid only if the collisions are very fast. This problem has been recently discussed by Horbatsch and Dreizler [25,263.
References
111F. Garcia Moliner and F. mores, Introduction
to the Theory of Solid Surfaces (Cambridge University Press, Cambridge, 1979). PI J.T. Devreese, I?. Brosens and L.F. Lemmens, Phys. Rev. B21 (1980) 1349. 131 N.R. A&a and W. Brandt, J. Phys. Cl6 (1983) L1217. and R.M. Ritchi~ Solid f41 P-H. Echenique, R. Nietien State Comm. 37 (1981) 779. PI E. Fermi and E. Teller, Phys. Rev. 72 (1947) 399. WI J. Lindhard and A. Winther, K. Dan. Vid. Selsk. Mat. Fys. Medd. 34 (l964) no. 4.
corrections
37
R. Piiz..s and Ph Nozieres, The Theory of Quantum Liquids, voi, 1 (Benjamin Press, New York, f%6) p. 322. D.N. Tripathy and S.S. MandaI, Phys. Rev. Bid (1977) 231. Yu. S. Sayasov, Z. Phys. A313 (1983) 9. L. Lantto, P. P&t&&en and A. Kallio, Phys. Rev. B26 (1982) 5565. S. Ichimaru and K. Atsumi, Phys. Rev. B24 (1981) 7385. N. Iwamota and D. Pines, Phys. Rev. B29 (1984) 3924, 3936. V.P. Silin, Zh. Eksp. ‘-&or. Fii 37 (1959) 273. L. Lantto, Pbys. Rev. B22 (1980) 1380. F.A. Stevens and H.A. P&rant, Phys. Rev. A8 (1973) 990. J. Pinnernan, Dissertation, The Institute of Physics, I$rhus University (1968) ~~p~b~~~. P. Sigmund, Phys. Rev. A26 (1982) 2497. F. Calogero, VariabIe Phase Approach to Potential Scattering (Academic Press, New York, 1967) p. 12. R. Blume, W. E&stein and H. Verbeek, Null. Instr. and Meth. 194 (1982) 67. D. Isaacson, New York University Dot. No. 02698 (National Auxiliary Publication Service, New York, 1975). R. Nunez, P.M. Eehenique and R.M. Ritchie, J. Phys. Cl3 (1980) 4229. R. Ray and G.D. Maha.n, Phys. Lett. 424 (1972) 301. P. Bauer, D. Semrad and R. Golser, Nucl. Instr. and Meth. B2 (1984) 149. J. Giber, I. Nagy, J. L&&b, Nucl. lnstr. and Meti. E2 (1984) 135. H. Horbatsch and R.M. Dreser, Z. Phys. A300 (1981) 119. H. Horbatscb and R.M, Dreizler, Z. Phys. A308 (1982) 329.
STOPPING PGWER WORKSHOP
34
LOCAL-FIELD CORRECTIONS FOR PROTONS
IN THE CALCULATION
OF ELECTRONIC
STOPPING
POWER
I. NAGY, J. LASZLb and J. GIBER Department
of Atomic Physics, Technical University Budapest, Budafoki w 8., H- 111 I Budapest, Hungary
Received 19 October 1984 and in revised form 14 December 1984
The energy loss of slow protons in a degenerate electron gas is calculated on the basis of the dielectric theory. The influence of various longitudinal dielectric functions is examined. Comparison with the results of the kinetic theory is made. The effect of the local-field correction on the straggling parameter is also dibsed.
1. Energy loss
a = (4/9n)1’3
The change of the kinetic energy (W) per unit path length of a proton (regarded as a point charge) moving in a dielectric medium and having velocity u at a certain point of the trajectory is given by the well known formula u-‘(dW/dt)=u-’
euE(r,
t)l_,,,,
(1)
where E(r, t) is the space and time dependent electric field strength generated by the moving charged particle. Formula (1) is equivalent to the one of Flores and Garcia Moliner [l] applying the induced potential u-‘(dW/dt)
= --u-l
e[a+(r,
t)/i3 t] (,=vt.
Imfdk
vk/e(nk,
k) k*,
(2)
where the quantity S introduced denotes the stopping power usually used in particle-solid interactions. E is the dielectric function of the medium, k is the wave vector. The dielectric function of a degenerate electron gas at absolute zero temperature is defined [2] as e( 0, k) = I + Q&l-
Q&h
(3)
where o = uk, Q, is Lindhard’s complex polarizability [6], G is the so called local-field correction. Both Qa and G depend on w and k. The dependence of G on o can be neglected if u QCur ( ur is the Fermi velocity) Introducing
the usual notations
x2 = e*/rhu,
= arJa,
0168-583X/85/$03.30 8 Elsevier Science Publishers (North-Holland Physics Publishing Division)
-KC(r,)u=
density
=L’dz
remains
the first Bohr
--ylu
can be obtained, where electron mass. Thus C(r,)
and a,
z’/{
(4) K = 4e4m%/3ah3,
z* -x2
to be determined,
A(z)
[G(z)
m,
- II}’
is the
(5)
where
ji(z)=1/2+(1-z2)(4z)-11n((z+l)/(z-1)]. (6) 2. Analysis of C(r,) 2.1. The case of G=O Several approximate forms are known in the literature. If the so called small-k approach for E is used in first order (i.e. ji = l), then C(r,)=
[ln(l
+1/X2)-(1
+x2)-*l/2
(7)
results. This shape, shown as curve 1 in fig. 1, is known [3,4] as the modification of the early Fermi-Teller formula C(r,) - ln(l/x*)[S]. If ji is expanded up to second order (j, = 1 - z*/3), the result is still an analytical formula C(r,)=9(3
-X2)-*{ln[(2 - (3 - x2)/(3
x*-t + 2x2)}/2
3)/3
x2] (8)
(curve 2 in fig. 1). Using the complete function ji the Lmdhard-Winther result [6] is obtained after numerical integration (curve 3 in fig. 1). It is obvious from the theory of the quantum-mechanical many-body problem,
PI. z = k/2k,,
S,=
n is the electron
(1’)
If the medium is homogeneous and infinite, and if the change of the field is slight enough (i.e. if Maxwell’s laws of linear electrodynamics can be used), then the following equation can be deduced from (1) after the Fourier transformation of the space variables S, = e* (2a*u)-1
where radius
and rsa, = (3/4~n)l’~,
B.V.
I. Nagv et al. / Local-field corrections
35
function of Silin [13] is applied to arrive at
C(rJ
‘I~ cl&J
C( rs) = (1 + Fi/3)(1
1.5 :
X [ln(l+
+ F:)-*
l/x:)-{1
+xI)-J/2,
(9)
which is formally very similar to (7). Here x;=xz(l+F;)-l
(l+F;/3)-‘.
According to Pines and Nozieres [7], Fz and F; can be determined from the effective mass formula mz/m,
= 1 -I- F[/3
and from the compressibility x0/x = (1+ F;),‘(l
0
i 0
. 0.5
1
Z2a0.166rS
Fig. 1. Various approximations for C( r*) functions (eq. (5)). Curve 1: f, = 1, G = 0. Curve 2: fi = 1- z*/3, G = 0. Curve 3: Lindhard-Winther with compiete fi, G = 0. Curve 4: complete fr, G(z) after Ikvreese et al. Curve 5: complete ft, G(z, r*) afte Lantto et al. Curve 6: results of density-functional formalism after Echenique et al. Circles: results of Landau theory of Fermi liquids for ‘; = 1,2, 3and4.
2.2. ThecaseofG+O.
2.3. Application of binary collision theory From the theory in which the proton-electron interaction is considered as a pure binary collision (Finneman) 1161,Sigmund [17]) m, eti(kr)u
(10)
can be obtained, where n is the gas density, transport cross-section and v SE ur. Applying n = k;/3n2, and
The exchange and correlation contributions are taken into account by the local-field correction function. The calculations of Devreese et al. [2] and Tripathy and Mandal [&] considering only the exchange cont~bution result in a universal G(z) function (i.e. not depending on rs). For small z values G(z) - z’, while at z z 1 (i.e. k = 2k,) G has a sharp maximum (G,, P 2). Using (5), this correction leads to a C(r,) curve (4 in fig. 1) with a minimum (contrary to the result of Sayasov [9]). This is unexpected from the physical point of view, since an electron gas of lower density should not slow down a proton more efficiently. According to theories that consider also the correlation [lo-121, G(z) has no such sharp maximum and G_ zs 1, if z = 1 (G(z) - z2 is still valid for small z values). Also, G shows a slight dependence on r,. Using the data of Lantto et al. [lo] for G(z, rS), our resulting C(r,) can also be seen fig. 1 (curve 5). At last a numerical calculation is shown, which is based on the Landau theory of Fermi liquids. The E
-t-F;),
where x0 and x are the ~mpressib~ties of the free and the correlated electron gas, respectively. For the theoretical determination of these quantities, the exact value of the correlation energy has to be known [14,15]. Using the data of Pines and Notieres [7] our corresponding I values from eq. (9) (for r, = 1,2, 3 and 4) are shown in fig. 1 by circles . These points fully agree with the function values calculated with the help of G(z, rS) after Lantto et al. [lo].
Sn = --n+ that the Lindhard function is only applicable if r, -=K1 E71.
ratio
vF = e2/nhX2,
au(kF)=*
f
(‘Itl)
a,, the
kF = rng&h
sin2(6r--SI+,)
k6 I-O in eq. (10) we obtain sa=
-KC’(r,)v
= -yr 0,
where C’(r,)=(~x~)-~~
(11) (I+l)
sin2 (al-a,,,)
with
8, being the phaseshifts’:fO the electron scattering on a screened proton and K is the same constant as in eq. (4). For this calculation knowledge of the scattering potential is necessary. Using the static (I(W+ 0, k) with fi = 1, the usual Yukawa (Debye) type screened potential can be obtained (in atomic units V(r) = 2 exp (-kor)/r, where k. = 2/(a’~~)~/*). In the first Born approximation with this potential one would obtain exactly formula (7), since ,go(f+
1) sin2@,-&+,)
= ( nX2)2[ ln(1 + l/x’)
- (1+ x2)-l]
/2.
STOPPING POWER WORKSHOP
I. Nagy et al. / Local-field corrections
36
This result is not surprising, as in the region where the Lindhard function can be used (r, -K 1) the Fermi velocity ur is high (i.e. ur 3* e2/h), and the Born approximation is good for high energies. The screened potential of the proton has been calculated self-consistently by Echenique et al. [4] within the density-functional formalism. Afterwards the shift factors are determined numerically [HI. This leads to curve 6 in fig. 1. It should be noted that in this case C’(r,) decreases more strongly with increasing r,. This is to be expected, since the induced screened potential is the result of a non-linear calculation exceeding the linear response theory. So as r, increases, the energy loss of the proton decreases more rapidly (within this theory) than is predicted by the linear theory. This is due to the occurrence of bound states of atomic character which tend to screen interactions with the electron gas.
deviation is of the order of a few percent. The various approximations to the Lindhard function (eq. (6)) which gave very similar curves for C( rs) now give considerably different functions C,(r+): curve 3 (corresponding to fr =I 1 - z2/3) or curve 4 (where fr = 1). Jn these latter cases C2( rs) can be expressed analytically.
The measurable energy change of a proton passing through a metal foil of thickness Ax is - AW = W, W( AX), where W, is the primary energy, W(Ax) is the transmitted particle energy. If formula (2) or (11) is considered a simple Newton equation of motion and the nuclear stopping is neglected, then mP t=
-yv
(13)
results. Thus, the kinetic energy after penetration depth Ax is
3. Straggling As could be seen above, the introduction of G(z) f 0 causes an essential change in the C(r,) curves in comparison to the result of the Lindhard-Winther theory. This is not so for the straggling parameter. According to Sigmund [17] the straggling parameter (T) in the low velocity range is
W(Ax)
(12) where the formula for C,(r,) is similar to eq. (5) but z4 must be substituted for z3. The calculation of C,( r,) with the complete f*(z) function (eq. (6)) results in curve 1 in fig. 2. Considering G(z, rs) after Lantto et al. [lo], we obtain curve 2 in fig. 2. It can be seen that the
C,(rs) i
uo)‘,
(14)
where va is the initial velocity. If J%$ is given in keV, and Ax in A, then the following simple formula can be obtained from (14) W(Ax)=
T= 4ae4$(u/u,)2C,(r,),
= W,(l - yAx/m,
into
Wo[l -2.2X1O-3
C(rs)AxfW~‘z]2
(14)
In the measurements of Blume et al. [19] the gold target was bombarded by protons in channeling directions, In this case r, = 1.49 (i.e. x2 = 0.247) [20]. Our results for C(r,) from eq. (9) (or from (5) using the local-field correction proposed by Lantto et al. [lo]) is C( r.) = 0.7. (According to the Lindhard theory this value would be 0.46.) Table 1 contains the calculated and experimental W(Ax) values for a pair of primary energies and target thicknesses. Our theoretical results show very good agreement with the measured data within experimental error. Some more physical effects should be taken into account for developing the theory. Such phenomena are the surface effects at the place of entrance and escape of the projectile (e.g. for metals, the surface plasmon generation) [21,22], the possible inhomogeneity of the elecTable 1 Experimental and theoretical transmitted particle energy valttes for gold target
0
4 0
e
0.5
1
?&0.166 rs
Fig. 2. Various approximations for C,( );) functions. Curve 1: L~dh~d-umber with complete _f,, G = 0. Curve 2: complete II, G(r, rS) after Lantto et al. Curve3:f,=l-r2/3, GiO. Curve4:f,=l,
G%O.
Target thickness Ax
[A]
620 1200
Primary energy
Transmitted particle energy -. W(Ax) [keVl
JGEkeVlExp.[19]
Theory (present)
Theory darned)
6 18
2.25 5.75
3.32 9.17
- 2.5 : 6
I. Nags et al. / Local-field
tron gas and moreoverT in the case of LDA (locai-density a~~ro~mation~ f23], the gradient of the electron density close to the lattice atoms. If the bombardment is done avoiding cbmeling directions, an additive speed change of the projectile takes place when passing ‘close to a target atom due to the Coulomb interaction [24]. In relation to the LDA theory we must add that the determination of the electron density induced by a given proton-solid atom interaction can be made only by the solution of the time dependent SchrBdinger equation. In the case of slow velocities, this problem leads to the quasistationary problem for a quasimolecub. The usual atomic form of the electron density of the target atom is valid only if the collisions are very fast. This problem has been recently discussed by Horbatsch and Dreizler [25,263.
References
111F. Garcia Moliner and F. mores, Introduction
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