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Asymmetry of self-energies of electrons and positrons in solids
14
Nuclear
ASYMMETRY
OF SELF-ENERGIES
Instruments
OF ELECTRONS
and Methods
in Physics Research
AND POSITRONS
B48 (1990) 14-17 North-Holland
IN SOLIDS
Shigeru SHINDO Tokyo Gakugei
University. Koganei,
Tokyo 184, Japan
We calculate the self-energies of positrons and electrons in solids. Our calculation shows that the imaginary part of the self-energy of positrons is much enhanced in comparison with that of electrons, which explains the recent experiments. The difference of the self-energies of positrons and electrons in solids is explained as the polarization effect of the solid atoms induced by the incident charges.
1. Introduction
The difference of the stopping power for positively and negatively charged particles was pointed out experimentally by Barkas et al. [l] and is now known as the Barkas effect. This effect is known as a higher-order perturbation effect for the stopping power [2-71. In a previous paper [7] we presented a theory of the stopping power by including the higher-order perturbation terms. There we developed a way to take into account the contributions of the close collisions to the Barkas effect. In this paper we apply the theory of the stopping power to the calculation of the self-energies of electrons and positrons in solids. A recent slow-positron surface scattering experiment [8] suggests that inelastic scatterings of positrons in solids are more important in comparison with those of electrons. The inelastic scattering cross section is represented by the imaginary part of the self-energy which is closely related to the stopping power. We have seen in a previous paper [7] that, for the stopping power, higher-order perturbation terms explain the asymmetry between positively and negatively charged particles. Then we investigate the contributions of the higher-order terms for the self-energies of positrons and electrons in solids.
2. Semiclassical harmonic-oscillator model In this section, we briefly summarize the semiclassical harmonic-oscillator model for the stopping power of solids for charged particles [7]. We assume that electrons of the target material are bound isotropically and harmonically to the nucleus. We also assume that the incident charged particle interacts with an electron in the material. In order to describe the collision process of the incident ion and target electron accurately, we choose this simplified atomic model, which was originally applied by Bohr [9] to his pioneering classical0168-583X/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
mechanical theory of stopping power and has been extended by various authors [2-71, to discuss the Barkas effect. Recently, Sigmund and Haagerup [lo] studied the stopping power in the limit of the first Born approximation by using this atomic model. According to the harmonic-oscillator model, the energy transfer (T) due to the collision is described by (T)
= ;J_”
cc/’-m(F(j)F(j’))[I
Xexp[io(t
+
n(a)1
- f’)] dt’ dt,
(I)
where m is the electron mass, w is the angular frequency of the oscillator, F(r) represents the Coulomb force acting between the incident charged particle and the target electron at time t, ( ) means the expectation value over the initial wave function of the incident particle, and n(w) is the average occupation number of the oscillator, which is negligible for the electronic excitation. We showed [7] that the energy transfer (T) can approximately be decomposed into a summation of the partial energy loss value T, such that (T)
= x
/
I)T,,
(2)
where (I + 1)n is an initial angular momentum of the center-of-mass coordinate system of the incident ion and the target electron. Now we restrict ourselves to a semiclassical calculation for the partial energy loss T, of eq. (2). In this case, (F(j)F(t’)) in eq. (1) is replaced by F(x,(t))F(x,(f’)), where x,(r) is the collision trajectory whose initial angular momentum L = pub is chosen as A( I + l), where b is the impact parameter, u the velocity of the charged particle and p the reduced mass of the electron and the incident charged particle. Then T, of eq. (2) is given by T,= (1/2m)ll_X_F(x,(j)) A higher-order
perturbation
exp(iwt) effect,
dri2.
(3)
that is the polari-
S. Shindo / Asymmetry
of self-energies of electrons and positrons in solids
zation of the target atom, is taken into account in the calculation of the collision trajectory x,(t). The stopping cross section S can be expressed by using T/ of eq. (3) as [7] S=2n(A//J”)ZC(I+1)~.
15
where F( t ) is the Coulomb force operator of eq. (1) and n (w ) is the occupation number which can be neglected for the electronic excitation. When we use the Born approximation, the self-energy (5) can be written as
(4)
In the previous paper [7], we have shown that the stopping cross section represented by eq. (4) contains higher-order corrections to the Bethe formula: the Barkas term and the Bloch correction term.
3. Self-energy of positrons and electrons
It has recently been recognized that a low-energy positron beam provides a powerful probe for studying solid surfaces such as low-energy positron diffraction (LEPD), positron energy loss spectroscopy (PELS), positronium formation spectroscopy (PsFS) [ll] and so on [12]. However, the cross sections of the inelastic scatterings of positron in solids have not been calculated till now. According to a recent slow-positron surface scattering experiment by Mayer et al. [8], inelastic scatterings of positrons in solids are more important in comparison with the electron beam. In the experimental analysis of the low-energy electron (positron) diffraction, contributions from the inelastic scattering are expressed in terms of the self-energy Z. The real part contains contributions from the virtual excitations and simply represents the additional contributions to the elastic scattering. The imaginary part represents absorption of the elastically scattered beams and is related with the mean free path of the inelastic scattering, X, by
(6) where ) k) and ) k’) are the initial and final plane-wave states, respectively. The self-energy in the first-order perturbation theory given by eq. (6) is proportional to the square of the incident charge. Then the experimental difference between the positron and electron self-energies cannot be explained by using this first-order theory. In our harmonic oscillator model, the self-energy 2 of eq. (5) can be related to the stopping cross section S of eqs. (1) and (4) as follows:
- -d(T) = snv = -2wp, dt where n is the electron imaginary part of the semiclassical trajectory section S of eq. (4), the -1 ,Xtm = _Vun2T 2w
x I<[+ I
density and self-energy. method for self-energy
Elm represents the When we use the the stopping cross becomes
A2 2m(pu)’ I)l/_mmF(x,(t))
exp(iwt)
dr12.
(8)
The real part of the self-energy ZRe is related to its imaginary part by the Kramers-Kronig’s relation:
2’” = Av/2X. From the experiment by Mayer et al. [8], Zlrn for positrons is determined to be -6 eV and that of the electron self-energy to be -4 eV for copper (the range of the incident energy is 50-450 ev). The self-energy is closely related to the stopping power which we discussed in the previous section, where we showed that higher-order perturbation terms explain the difference of the stopping power for positively and negatively charged particles. Therefore we apply the previous theory of the stopping power to the self-energy and we show that the difference of the self-energy of electrons and positrons in solids is originated from the higherorder perturbation effect. Using the harmonic oscillator model, the self-energy can be represented in the Heisenberg picture as follows [13]:
xexp[iw(t-
t’)] dt’,
For the numerical calculation of eq. (8), x,(t) is determined from the classical equations of motion [7],
c,(t)
X
/
’ w sin[w(t-t’)]F(x,(t’)) --1o
dt’,
(10)
the initial conditions are chosen as x,( t = 0) = t2(l+l)/mv), a,(r=O)=(v,O). It is worth to note that eqs. (8) and (9) coincide with those given from the first Born approximation of eq. (6) when we choose the straight path for the collision trajectory, f,(t) = (v, 0). Therefore higher-order perturbation effects are contained in the F-dependence of the collision trajectory, x,(t), determined from eq. (10). To adapt our harmonic-oscillator model for real atoms, the self-energy Z should be averaged over the frequency distribution F(w) such as [14] where
(-co,
8= (5)
= F(x,(t)) - (P/m)
s
dwF(w)Z(w),
(II)
I. EXCITATION.
STOPPING
S. Shindo /Asymmetry
16
of self-energies
of electrons and positrons in solid? ?(a.u.)
a
6
0
1
I
I
4
k
I
I
I
I
1
23456789
I
I
I
23456789
v(a.u.1
I
I
1
1
I
v(a. u.)
Fig. 1. Self-energies of electrons and positrons in copper as a function of the velocity. The line marked by e+ is calculated for positrons, and that by e- for electrons. (a) Imaginary part of the self-energy. Horizontal lines are the experiments by Mayer et al. [8] for electrons and positrons. (b) Real part of the self-energy.
where X(w) is given by eqs. (8) and (9). Using the Thomas-Fermi statistical atomic model and the local plasmon approximation, r(w) is calculated by
This explains the experimental mean inner potential of positrons, determined to be 0 eV by Mayer et al. [8].
r(bJ)=/+-
4. Concluding remarks
up(p) =
x~,(P)]n(p) dp,
[4an(p)e2/m]1’2,
where n(p) is the electronic density of the target atom at the point p, x is a fitting parameter chosen as x = fi [14], and n in eq. (8) is the electron density of the target atom at the point p, n = n(p) In fig. 1, we show the calculation of the positron and electron self-energies in copper as a function of the velocity. Fig. la represents the imaginary and fig. lb the real part, respectively. In fig. la it can be seen that the imaginary part is much enhanced for the positron. In fig. la the horizontal line indicates the experimental value in this energy range. A qualitative agreement is obtained for the electron and positron self-energies. We state that the difference between the positron and electron self-energies is explained from our model which includes the polarization effect of the target atom induced by the incident charge. The real parts of the self-energies of electrons and positrons are of the same order in the whole velocity range. However, the magnitude of the mean inner potential, given by (V) + BRe, where (I’) is the average potential in solids, differs for electrons and positrons because (I’) is positive for positrons while it is negative for electrons. Since BRe is a negative quantity for electrons and positrons, the magnitude of the mean imrer potential (V) + ERe of positrons is less in comparison with that of electrons.
We developed a theory of the self-energy of positrons and electrons in solids, by using the semiclassical harmonic-oscillator model. The first-order perturbation theory of the self-energy can be improved by using our theory. We compared our calculation with experiments. It is shown that our theory agrees, at least qualitatively, with the experiments. We note that our theory is essentially based on the high-velocity approximation. The experimental condition is in the low- and intermediatevelocity range. Therefore we should develop the theory of the stopping power and the self-energy of low-velocity ions by including the higher-order terms in a following work. The author would like to thank Drs. A. Ishii and H. Nitta for valuable discussions. This work was performed with use of the HITAC M-680H/S-820 computer at the Institute of Molecular Science, Okazaki, Japan.
References [l] W.H. Barkas, N.J. Byer and H.H. Heckman, Phys. Rev. Lett. 11 (1963) 26. [2] J.C. Ashley, W. Brandt and R.H. Ritchie, Phys. Rev. B5 (1972) 2393.
S. Shindo / Asymmetry
of self-energies of electrons
[3] J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. A8 (1973) 2402. [4] J.D. Jackson and R.L. MacCarthey, Phys. Rev. B6 (1974) 4131. [5] K.W. Hill and E. Merzbacher, Phys. Rev. A9 (1974) 156. [6] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [7] S. Shindo and H. Minowa, Phys. Status Solidi B145 (1988) 89. [8] R. Mayer, C.-S. Zhang, K.G. Lynn, W.E. Frieze, F. Jena and P.M. Marcus, Phys. Rev. B35 (1987) 3102. [9] N. Bohr, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 18, no. 8 (1948) 1.
17
and positrons in soliak
[lo] P. Sigmund and U. Haagerup, Phys. Rev. A34 (1986) 892. [ll] B. Pendry, in: Positron Solid State Physics, eds. W. Brandt and D. Dupasquire (North-Holland, New York, 1983) p. 432. [12] A.P. Mills, Jr., L. Pfeiffer and P.M. Platzman, Phys. Rev. Lett. 51 (1983) 1085. [13] S. Shindo, Surf. Sci. 159 (1985) 283. [14] J. Lindhard and M. Sharff, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 27, no. 8 (1953) 15.
I. EXCITATION,
STOPPING
Nuclear
ASYMMETRY
OF SELF-ENERGIES
Instruments
OF ELECTRONS
and Methods
in Physics Research
AND POSITRONS
B48 (1990) 14-17 North-Holland
IN SOLIDS
Shigeru SHINDO Tokyo Gakugei
University. Koganei,
Tokyo 184, Japan
We calculate the self-energies of positrons and electrons in solids. Our calculation shows that the imaginary part of the self-energy of positrons is much enhanced in comparison with that of electrons, which explains the recent experiments. The difference of the self-energies of positrons and electrons in solids is explained as the polarization effect of the solid atoms induced by the incident charges.
1. Introduction
The difference of the stopping power for positively and negatively charged particles was pointed out experimentally by Barkas et al. [l] and is now known as the Barkas effect. This effect is known as a higher-order perturbation effect for the stopping power [2-71. In a previous paper [7] we presented a theory of the stopping power by including the higher-order perturbation terms. There we developed a way to take into account the contributions of the close collisions to the Barkas effect. In this paper we apply the theory of the stopping power to the calculation of the self-energies of electrons and positrons in solids. A recent slow-positron surface scattering experiment [8] suggests that inelastic scatterings of positrons in solids are more important in comparison with those of electrons. The inelastic scattering cross section is represented by the imaginary part of the self-energy which is closely related to the stopping power. We have seen in a previous paper [7] that, for the stopping power, higher-order perturbation terms explain the asymmetry between positively and negatively charged particles. Then we investigate the contributions of the higher-order terms for the self-energies of positrons and electrons in solids.
2. Semiclassical harmonic-oscillator model In this section, we briefly summarize the semiclassical harmonic-oscillator model for the stopping power of solids for charged particles [7]. We assume that electrons of the target material are bound isotropically and harmonically to the nucleus. We also assume that the incident charged particle interacts with an electron in the material. In order to describe the collision process of the incident ion and target electron accurately, we choose this simplified atomic model, which was originally applied by Bohr [9] to his pioneering classical0168-583X/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
mechanical theory of stopping power and has been extended by various authors [2-71, to discuss the Barkas effect. Recently, Sigmund and Haagerup [lo] studied the stopping power in the limit of the first Born approximation by using this atomic model. According to the harmonic-oscillator model, the energy transfer (T) due to the collision is described by (T)
= ;J_”
cc/’-m(F(j)F(j’))[I
Xexp[io(t
+
n(a)1
- f’)] dt’ dt,
(I)
where m is the electron mass, w is the angular frequency of the oscillator, F(r) represents the Coulomb force acting between the incident charged particle and the target electron at time t, ( ) means the expectation value over the initial wave function of the incident particle, and n(w) is the average occupation number of the oscillator, which is negligible for the electronic excitation. We showed [7] that the energy transfer (T) can approximately be decomposed into a summation of the partial energy loss value T, such that (T)
= x
/
I)T,,
(2)
where (I + 1)n is an initial angular momentum of the center-of-mass coordinate system of the incident ion and the target electron. Now we restrict ourselves to a semiclassical calculation for the partial energy loss T, of eq. (2). In this case, (F(j)F(t’)) in eq. (1) is replaced by F(x,(t))F(x,(f’)), where x,(r) is the collision trajectory whose initial angular momentum L = pub is chosen as A( I + l), where b is the impact parameter, u the velocity of the charged particle and p the reduced mass of the electron and the incident charged particle. Then T, of eq. (2) is given by T,= (1/2m)ll_X_F(x,(j)) A higher-order
perturbation
exp(iwt) effect,
dri2.
(3)
that is the polari-
S. Shindo / Asymmetry
of self-energies of electrons and positrons in solids
zation of the target atom, is taken into account in the calculation of the collision trajectory x,(t). The stopping cross section S can be expressed by using T/ of eq. (3) as [7] S=2n(A//J”)ZC(I+1)~.
15
where F( t ) is the Coulomb force operator of eq. (1) and n (w ) is the occupation number which can be neglected for the electronic excitation. When we use the Born approximation, the self-energy (5) can be written as
(4)
In the previous paper [7], we have shown that the stopping cross section represented by eq. (4) contains higher-order corrections to the Bethe formula: the Barkas term and the Bloch correction term.
3. Self-energy of positrons and electrons
It has recently been recognized that a low-energy positron beam provides a powerful probe for studying solid surfaces such as low-energy positron diffraction (LEPD), positron energy loss spectroscopy (PELS), positronium formation spectroscopy (PsFS) [ll] and so on [12]. However, the cross sections of the inelastic scatterings of positron in solids have not been calculated till now. According to a recent slow-positron surface scattering experiment by Mayer et al. [8], inelastic scatterings of positrons in solids are more important in comparison with the electron beam. In the experimental analysis of the low-energy electron (positron) diffraction, contributions from the inelastic scattering are expressed in terms of the self-energy Z. The real part contains contributions from the virtual excitations and simply represents the additional contributions to the elastic scattering. The imaginary part represents absorption of the elastically scattered beams and is related with the mean free path of the inelastic scattering, X, by
(6) where ) k) and ) k’) are the initial and final plane-wave states, respectively. The self-energy in the first-order perturbation theory given by eq. (6) is proportional to the square of the incident charge. Then the experimental difference between the positron and electron self-energies cannot be explained by using this first-order theory. In our harmonic oscillator model, the self-energy 2 of eq. (5) can be related to the stopping cross section S of eqs. (1) and (4) as follows:
- -d(T) = snv = -2wp, dt where n is the electron imaginary part of the semiclassical trajectory section S of eq. (4), the -1 ,Xtm = _Vun2T 2w
x I<[+ I
density and self-energy. method for self-energy
Elm represents the When we use the the stopping cross becomes
A2 2m(pu)’ I)l/_mmF(x,(t))
exp(iwt)
dr12.
(8)
The real part of the self-energy ZRe is related to its imaginary part by the Kramers-Kronig’s relation:
2’” = Av/2X. From the experiment by Mayer et al. [8], Zlrn for positrons is determined to be -6 eV and that of the electron self-energy to be -4 eV for copper (the range of the incident energy is 50-450 ev). The self-energy is closely related to the stopping power which we discussed in the previous section, where we showed that higher-order perturbation terms explain the difference of the stopping power for positively and negatively charged particles. Therefore we apply the previous theory of the stopping power to the self-energy and we show that the difference of the self-energy of electrons and positrons in solids is originated from the higherorder perturbation effect. Using the harmonic oscillator model, the self-energy can be represented in the Heisenberg picture as follows [13]:
xexp[iw(t-
t’)] dt’,
For the numerical calculation of eq. (8), x,(t) is determined from the classical equations of motion [7],
c,(t)
X
/
’ w sin[w(t-t’)]F(x,(t’)) --1o
dt’,
(10)
the initial conditions are chosen as x,( t = 0) = t2(l+l)/mv), a,(r=O)=(v,O). It is worth to note that eqs. (8) and (9) coincide with those given from the first Born approximation of eq. (6) when we choose the straight path for the collision trajectory, f,(t) = (v, 0). Therefore higher-order perturbation effects are contained in the F-dependence of the collision trajectory, x,(t), determined from eq. (10). To adapt our harmonic-oscillator model for real atoms, the self-energy Z should be averaged over the frequency distribution F(w) such as [14] where
(-co,
8= (5)
= F(x,(t)) - (P/m)
s
dwF(w)Z(w),
(II)
I. EXCITATION.
STOPPING
S. Shindo /Asymmetry
16
of self-energies
of electrons and positrons in solid? ?(a.u.)
a
6
0
1
I
I
4
k
I
I
I
I
1
23456789
I
I
I
23456789
v(a.u.1
I
I
1
1
I
v(a. u.)
Fig. 1. Self-energies of electrons and positrons in copper as a function of the velocity. The line marked by e+ is calculated for positrons, and that by e- for electrons. (a) Imaginary part of the self-energy. Horizontal lines are the experiments by Mayer et al. [8] for electrons and positrons. (b) Real part of the self-energy.
where X(w) is given by eqs. (8) and (9). Using the Thomas-Fermi statistical atomic model and the local plasmon approximation, r(w) is calculated by
This explains the experimental mean inner potential of positrons, determined to be 0 eV by Mayer et al. [8].
r(bJ)=/+-
4. Concluding remarks
up(p) =
x~,(P)]n(p) dp,
[4an(p)e2/m]1’2,
where n(p) is the electronic density of the target atom at the point p, x is a fitting parameter chosen as x = fi [14], and n in eq. (8) is the electron density of the target atom at the point p, n = n(p) In fig. 1, we show the calculation of the positron and electron self-energies in copper as a function of the velocity. Fig. la represents the imaginary and fig. lb the real part, respectively. In fig. la it can be seen that the imaginary part is much enhanced for the positron. In fig. la the horizontal line indicates the experimental value in this energy range. A qualitative agreement is obtained for the electron and positron self-energies. We state that the difference between the positron and electron self-energies is explained from our model which includes the polarization effect of the target atom induced by the incident charge. The real parts of the self-energies of electrons and positrons are of the same order in the whole velocity range. However, the magnitude of the mean inner potential, given by (V) + BRe, where (I’) is the average potential in solids, differs for electrons and positrons because (I’) is positive for positrons while it is negative for electrons. Since BRe is a negative quantity for electrons and positrons, the magnitude of the mean imrer potential (V) + ERe of positrons is less in comparison with that of electrons.
We developed a theory of the self-energy of positrons and electrons in solids, by using the semiclassical harmonic-oscillator model. The first-order perturbation theory of the self-energy can be improved by using our theory. We compared our calculation with experiments. It is shown that our theory agrees, at least qualitatively, with the experiments. We note that our theory is essentially based on the high-velocity approximation. The experimental condition is in the low- and intermediatevelocity range. Therefore we should develop the theory of the stopping power and the self-energy of low-velocity ions by including the higher-order terms in a following work. The author would like to thank Drs. A. Ishii and H. Nitta for valuable discussions. This work was performed with use of the HITAC M-680H/S-820 computer at the Institute of Molecular Science, Okazaki, Japan.
References [l] W.H. Barkas, N.J. Byer and H.H. Heckman, Phys. Rev. Lett. 11 (1963) 26. [2] J.C. Ashley, W. Brandt and R.H. Ritchie, Phys. Rev. B5 (1972) 2393.
S. Shindo / Asymmetry
of self-energies of electrons
[3] J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. A8 (1973) 2402. [4] J.D. Jackson and R.L. MacCarthey, Phys. Rev. B6 (1974) 4131. [5] K.W. Hill and E. Merzbacher, Phys. Rev. A9 (1974) 156. [6] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [7] S. Shindo and H. Minowa, Phys. Status Solidi B145 (1988) 89. [8] R. Mayer, C.-S. Zhang, K.G. Lynn, W.E. Frieze, F. Jena and P.M. Marcus, Phys. Rev. B35 (1987) 3102. [9] N. Bohr, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 18, no. 8 (1948) 1.
17
and positrons in soliak
[lo] P. Sigmund and U. Haagerup, Phys. Rev. A34 (1986) 892. [ll] B. Pendry, in: Positron Solid State Physics, eds. W. Brandt and D. Dupasquire (North-Holland, New York, 1983) p. 432. [12] A.P. Mills, Jr., L. Pfeiffer and P.M. Platzman, Phys. Rev. Lett. 51 (1983) 1085. [13] S. Shindo, Surf. Sci. 159 (1985) 283. [14] J. Lindhard and M. Sharff, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 27, no. 8 (1953) 15.
I. EXCITATION,
STOPPING