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Influence of electron exchange and correlation on the stopping power for slow ions in metals
0038-1098/82] 180437-03 $03.00/0 Pergamon Press Ltd.
Solid State Communications, Vol. 42, No. 6, pp. 437--439, 1982. Printed in Great Britain.
INFLUENCE OF ELECTRON EXCHANGE AND CORRELATION ON THE STOPPING POWER FOR SLOW IONS IN METALS M.D. Friedberg and S.A. Trigger Institute for High Temperatures, U.S.S.R. Academy of Sciences, 127412 Korovinskoye chausse, Moscow, U.S.S.R.
(Received 28 October 1981 by G.S. Zhdanov) The exchange and correlations between free electrons in metals are shown always to increase electronic stopping power for slow ions. The stopping power is expressed analytically for theoretical evaluations. This expression was used to calculate stopping powers for slow protons in Al and Ag. A better agreement with experimental data is noticed. 1. STOPPING OF SLOW, v < VF, ions in metals is usually treated in Born approximation [1,2]. Although the perturbation theory parameter, e2/hve is of order of unity for metallic densities, using of the lowest order of perturbation theory leads nevertheless to the straight proportionality between energy losses on unit path length and the velocity vp of projectile which is experimentally observed. A more general t-matrix approach consideration [3 ] confirms the validity of Born approximation calculations. The proportionality coefficient is not, however, satisfactory evaluated by perturbation theory approach as yet, and there exists a significant discrepancy of order of the factor 2-6 between its experimental and theoretical values. This discrepancy holds irrespectively of whether calculations are based on the approximate Ritchie [2] formula
Zv, vv,M v are the charge of the projectile, its velocity and mass, respectively. V~e = 41rh2e21q2; V~p = Zp v~e; x = cos (v~q); (l(q, eoa) is the polarization operator, which cannot be calculated in general form. ~(q, 60) = 1 -- V~efl(q, co) is the dielectric function of electron system. Method of obtaining equation (2) implies no restrictions on the type of interactions between electrons. Hence ~(q, co) is assumed to be an explicit dielectric function of electronic Fermi-liquid. In RPA stopping power may be calculated approximately in the analytical form, from equation (2), using the RPA-expression l]0(q, co) for l](q, co) [7]. Hence Vp < VF, we get h~q ,< eF and then use l]o(q, 0) instead of l~lo(q, wq). A following inequality (3) holds simultaneously I Im co(q, coq)l 2 < IRe co(q, wq)l 2.
- 3--na~Z~v, In
+
+(-(-~r,)|'
(1)
or whether they are carried by numerical methods [4, 5]. These results [2, 4, 5] are obtained within the framework of random-phase approximation (RPA) for electrons in metal. Polarizability of electronic system in RPA coincides with that of degenerate ideal electron gas. Electrons in metal are known to constitute strong interacting Fermi-liquid. The purpose of present investigation is to account for exchange and correlation of metallic electrons in calculations of stopping power.
(3)
We, further, replace exact expression for Re e0(q, 0) by longwave asymptotic Re co(q, 0)
> I + q~:F/q 2. q "-~ 0
After these approximating procedures integration in (2) can be performed with elementary functions; that gives Ritchie formula (1). The well known Fermi-Teller formula results from equation (1) as limiting case, corresponding r s < 1. 3. To account for exchange-correlation effects on an electronic stopping power one may use a polarization operator H(q, co) in the form [8].
2. Stopping power of electron gas is firstly formulated in [6], using diagram technique for Green function of many-body system
l'I = I]o1(1 + V~eG(q)flo).
dE _ 1 |i'd3q hc%lV~Pl 2 Im (I(q, coq) dt (2n'h) a . l1 -- Z~e[l(q, ~aq)l 2"
(2) 437
(4)
G(q) contains all the information about the interaction between electrons in electron liquid of metal. If explicit ~(q, w) is expressed with the help of equation (4), the same inequality (3) holds now for real and imaginary parts of ~, as well as other assumptions made in the
438
STOPPING POWER FOR SLOW IONS IN METALS
Vol. 42, No. 6
-dE eV 2
~
-a-i 'K
p~ +q/2
pF-q/2 -pF+q/~
2PF
q
01
Fig. 1. A diagram of domains of determination for Im eRPA: 1. Im eRPA = Or/2Xq~r/q2.)~6o/qvp); 2. Im eRPA = 0; 3. Im e_RP~= -- Qr/4XqTF/q~)(PF/q)
x {I --
2°-• |dx
vqe[l
[V(~Pt~
Im I] o --G(q)] Re I+Io}'
Bx
3, = I + ~ ; l+x
dA~(ev} ~
20
5.0 Up,[O8 / ~
1
I0
°° l •
•
•
(5)
We use the coordinates (q, rnvpx) while performing integration in equation (5), and the integration domain is shown in Fig. I for the case of slow projectiles. The diagram of domains of definition is unchangable when you use either Fermi-liquid polarization operator (4), or that in RFA [7]. The reason for it is that G(q) > 0 for all q, and it has no points of breaking the continuity. From equation (5) one may conclude, that exchange and correlations always increase stopping power, as soon as G(q) > 0 for all q. There has been published a great number of papers, dealing with properties of G(q) [9, I 0]. The main purpose of these papers was to correctly describe compressibility, pair correlation function and other properties of electron liquid at metallic densities, where r e = 2 - 6 . Here we use a simple parametrized formula for G(q) by Hedin and Lundqvist [11l.
a(q) = i ~ ( q / p F ) ~,
I0 u r
-
process of obtaining equation (1), do. Energy losses of projectile on unit path length are given by the following expression (2rth) -a f daq hwq
05
Fig. 2. Stopping power of Ag for slow protons [5]: Dots experiment; 1. Numerical calculations in RPA; 2. Analytical evaluation by Ritchie [2] ; 3. Microscopical theory of Fermi-liquid correlations [ 11 ], long-wave asymptotic for Re Ho.
~1 -- [(mvpx -- q/2)2/p~ ]}.
dE dx
0.2
(6) re
A
The advantage of this form of G(q) is that it allows the integration in equation (5) to be performed in general form. The procedure of obtaining the integral includes some approximations: explicit domain of integration is replaced by a rectangular restricted with point B. Im I] o is always taken in the form, corresponding domain 1,
Fig. 3. Stopping power of A1 for stow protons [5]; Dots - experiment; 1. Numerical calculations in RPA; 2. Formula by Ritchie; 3. Microscopical theory of Fermi-liquid correlations [ 11 ], long-wave asymptotic for Re H o. Fig. 1. The uncorrectness of result is small when the following formula
vp ,~ VF. Integration leads to
dE 2 ~2 (lr/ars)2 dx - 31r --',oZ~VPOr~erre _ .),)2 {
x ln[l+~r/ctr e-7]
lr/°tre--7 ) l+lr/ar s - 7 "
(7)
Expression (7) becomes equation (1) when 7 -~ 0. Some calculations are performed here, using A = 21 and B = 0.7734 [ 11 ], for conditions in Ag and A1. Values A and B are determined in [11 ] so as to satisfy calculations of electron liquid energy, performed in [12]. Figures 2 and 3 show plots of stopping power of Ag and A1 vs projectile velocity; experimental, theoretical RPA, theoretical for Fermi-liquid dielectric function. Fitting of theoretical to experimental values may be found essentially closer when electron exchange and correlations are taken into account.
Vol. 42, No. 6
STOPPING POWER FOR SLOW IONS IN METALS
4. Besides analytical evaluations of stopping power (5), a numerical calculation may be performed. One may find from [4, 5] that in RPA numerical calculations lead to stopping power values, which are greater than analytical evaluation (1). An analogous effect will take place when electrons in metal are treated as Fermi-liquid. Corresponding plots, analogous to those in Figs. 2 and 3 will constitute lines, situated above calculated with the help of equation (7). An accuracy of theoretical calculations of stopping power for slow ions in metals is assumed to increase. REFERENCES 1.
E. Fermi & E. Teller, Phys. Rev. 72,399 (1947).
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
439
R.H. Pdtchie,Phys. Rev. 114,644 (1959). T.L. Ferrell& R.H. Pdtchie,Phys. Rev. B16,115 (1977). G.J. lafrate & J.F. Ziegler, J. AppL Phys. 50, 5579 (1979). Yu.N. Yavlinsky, JETP80, 1622 (1981). A.I. Larkin, JETP37, 264 (1959). J. Lindhard, Danske vMensk, selsk. Mat. - Fys. 28, (1954). L.J. Sham, Phys. Rev. B7, 4357 (1973). P. Vashishta & K.S. Singwi,Phys. Rev. B6,876 (1972). D. Ceperely,Phys. Rev. B18, 3126 (1978). L. Hedin & B.I. Lundqvist, J. Phys. C." Solid StatePhys. 4, 2064 (1971). K.S. Singwi et al.,Phys. Rev. BI, 1044 (1970).
Solid State Communications, Vol. 42, No. 6, pp. 437--439, 1982. Printed in Great Britain.
INFLUENCE OF ELECTRON EXCHANGE AND CORRELATION ON THE STOPPING POWER FOR SLOW IONS IN METALS M.D. Friedberg and S.A. Trigger Institute for High Temperatures, U.S.S.R. Academy of Sciences, 127412 Korovinskoye chausse, Moscow, U.S.S.R.
(Received 28 October 1981 by G.S. Zhdanov) The exchange and correlations between free electrons in metals are shown always to increase electronic stopping power for slow ions. The stopping power is expressed analytically for theoretical evaluations. This expression was used to calculate stopping powers for slow protons in Al and Ag. A better agreement with experimental data is noticed. 1. STOPPING OF SLOW, v < VF, ions in metals is usually treated in Born approximation [1,2]. Although the perturbation theory parameter, e2/hve is of order of unity for metallic densities, using of the lowest order of perturbation theory leads nevertheless to the straight proportionality between energy losses on unit path length and the velocity vp of projectile which is experimentally observed. A more general t-matrix approach consideration [3 ] confirms the validity of Born approximation calculations. The proportionality coefficient is not, however, satisfactory evaluated by perturbation theory approach as yet, and there exists a significant discrepancy of order of the factor 2-6 between its experimental and theoretical values. This discrepancy holds irrespectively of whether calculations are based on the approximate Ritchie [2] formula
Zv, vv,M v are the charge of the projectile, its velocity and mass, respectively. V~e = 41rh2e21q2; V~p = Zp v~e; x = cos (v~q); (l(q, eoa) is the polarization operator, which cannot be calculated in general form. ~(q, 60) = 1 -- V~efl(q, co) is the dielectric function of electron system. Method of obtaining equation (2) implies no restrictions on the type of interactions between electrons. Hence ~(q, co) is assumed to be an explicit dielectric function of electronic Fermi-liquid. In RPA stopping power may be calculated approximately in the analytical form, from equation (2), using the RPA-expression l]0(q, co) for l](q, co) [7]. Hence Vp < VF, we get h~q ,< eF and then use l]o(q, 0) instead of l~lo(q, wq). A following inequality (3) holds simultaneously I Im co(q, coq)l 2 < IRe co(q, wq)l 2.
- 3--na~Z~v, In
+
+(-(-~r,)|'
(1)
or whether they are carried by numerical methods [4, 5]. These results [2, 4, 5] are obtained within the framework of random-phase approximation (RPA) for electrons in metal. Polarizability of electronic system in RPA coincides with that of degenerate ideal electron gas. Electrons in metal are known to constitute strong interacting Fermi-liquid. The purpose of present investigation is to account for exchange and correlation of metallic electrons in calculations of stopping power.
(3)
We, further, replace exact expression for Re e0(q, 0) by longwave asymptotic Re co(q, 0)
> I + q~:F/q 2. q "-~ 0
After these approximating procedures integration in (2) can be performed with elementary functions; that gives Ritchie formula (1). The well known Fermi-Teller formula results from equation (1) as limiting case, corresponding r s < 1. 3. To account for exchange-correlation effects on an electronic stopping power one may use a polarization operator H(q, co) in the form [8].
2. Stopping power of electron gas is firstly formulated in [6], using diagram technique for Green function of many-body system
l'I = I]o1(1 + V~eG(q)flo).
dE _ 1 |i'd3q hc%lV~Pl 2 Im (I(q, coq) dt (2n'h) a . l1 -- Z~e[l(q, ~aq)l 2"
(2) 437
(4)
G(q) contains all the information about the interaction between electrons in electron liquid of metal. If explicit ~(q, w) is expressed with the help of equation (4), the same inequality (3) holds now for real and imaginary parts of ~, as well as other assumptions made in the
438
STOPPING POWER FOR SLOW IONS IN METALS
Vol. 42, No. 6
-dE eV 2
~
-a-i 'K
p~ +q/2
pF-q/2 -pF+q/~
2PF
q
01
Fig. 1. A diagram of domains of determination for Im eRPA: 1. Im eRPA = Or/2Xq~r/q2.)~6o/qvp); 2. Im eRPA = 0; 3. Im e_RP~= -- Qr/4XqTF/q~)(PF/q)
x {I --
2°-• |dx
vqe[l
[V(~Pt~
Im I] o --G(q)] Re I+Io}'
Bx
3, = I + ~ ; l+x
dA~(ev} ~
20
5.0 Up,[O8 / ~
1
I0
°° l •
•
•
(5)
We use the coordinates (q, rnvpx) while performing integration in equation (5), and the integration domain is shown in Fig. I for the case of slow projectiles. The diagram of domains of definition is unchangable when you use either Fermi-liquid polarization operator (4), or that in RFA [7]. The reason for it is that G(q) > 0 for all q, and it has no points of breaking the continuity. From equation (5) one may conclude, that exchange and correlations always increase stopping power, as soon as G(q) > 0 for all q. There has been published a great number of papers, dealing with properties of G(q) [9, I 0]. The main purpose of these papers was to correctly describe compressibility, pair correlation function and other properties of electron liquid at metallic densities, where r e = 2 - 6 . Here we use a simple parametrized formula for G(q) by Hedin and Lundqvist [11l.
a(q) = i ~ ( q / p F ) ~,
I0 u r
-
process of obtaining equation (1), do. Energy losses of projectile on unit path length are given by the following expression (2rth) -a f daq hwq
05
Fig. 2. Stopping power of Ag for slow protons [5]: Dots experiment; 1. Numerical calculations in RPA; 2. Analytical evaluation by Ritchie [2] ; 3. Microscopical theory of Fermi-liquid correlations [ 11 ], long-wave asymptotic for Re Ho.
~1 -- [(mvpx -- q/2)2/p~ ]}.
dE dx
0.2
(6) re
A
The advantage of this form of G(q) is that it allows the integration in equation (5) to be performed in general form. The procedure of obtaining the integral includes some approximations: explicit domain of integration is replaced by a rectangular restricted with point B. Im I] o is always taken in the form, corresponding domain 1,
Fig. 3. Stopping power of A1 for stow protons [5]; Dots - experiment; 1. Numerical calculations in RPA; 2. Formula by Ritchie; 3. Microscopical theory of Fermi-liquid correlations [ 11 ], long-wave asymptotic for Re H o. Fig. 1. The uncorrectness of result is small when the following formula
vp ,~ VF. Integration leads to
dE 2 ~2 (lr/ars)2 dx - 31r --',oZ~VPOr~erre _ .),)2 {
x ln[l+~r/ctr e-7]
lr/°tre--7 ) l+lr/ar s - 7 "
(7)
Expression (7) becomes equation (1) when 7 -~ 0. Some calculations are performed here, using A = 21 and B = 0.7734 [ 11 ], for conditions in Ag and A1. Values A and B are determined in [11 ] so as to satisfy calculations of electron liquid energy, performed in [12]. Figures 2 and 3 show plots of stopping power of Ag and A1 vs projectile velocity; experimental, theoretical RPA, theoretical for Fermi-liquid dielectric function. Fitting of theoretical to experimental values may be found essentially closer when electron exchange and correlations are taken into account.
Vol. 42, No. 6
STOPPING POWER FOR SLOW IONS IN METALS
4. Besides analytical evaluations of stopping power (5), a numerical calculation may be performed. One may find from [4, 5] that in RPA numerical calculations lead to stopping power values, which are greater than analytical evaluation (1). An analogous effect will take place when electrons in metal are treated as Fermi-liquid. Corresponding plots, analogous to those in Figs. 2 and 3 will constitute lines, situated above calculated with the help of equation (7). An accuracy of theoretical calculations of stopping power for slow ions in metals is assumed to increase. REFERENCES 1.
E. Fermi & E. Teller, Phys. Rev. 72,399 (1947).
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
439
R.H. Pdtchie,Phys. Rev. 114,644 (1959). T.L. Ferrell& R.H. Pdtchie,Phys. Rev. B16,115 (1977). G.J. lafrate & J.F. Ziegler, J. AppL Phys. 50, 5579 (1979). Yu.N. Yavlinsky, JETP80, 1622 (1981). A.I. Larkin, JETP37, 264 (1959). J. Lindhard, Danske vMensk, selsk. Mat. - Fys. 28, (1954). L.J. Sham, Phys. Rev. B7, 4357 (1973). P. Vashishta & K.S. Singwi,Phys. Rev. B6,876 (1972). D. Ceperely,Phys. Rev. B18, 3126 (1978). L. Hedin & B.I. Lundqvist, J. Phys. C." Solid StatePhys. 4, 2064 (1971). K.S. Singwi et al.,Phys. Rev. BI, 1044 (1970).