- Email: [email protected]
Semiclassical model for δ-electron emission from multielectronic atoms
Physics Letters A 173 ( 1993 ) 469-472 North-Holland
PHYSICS LETTERS A
Semiclassical model for 8-electron emission from multielectronic atoms S.A. Gerasimov Department of Nuclear Physics,Instituteof Physics, RostovState University,Rostov-on-Don344104, RussianFederation Received 2 September 1992; revised manuscript received 23 December 1992;accepted for publication 29 December 1992 Communicated by L.J. Sham
A new semiclassical simple model is presented to study electron emission by atoms during heavy charged particle collisions. This model is based on the use of the plane wave Born theory and the semiclassical WKB approximation for atoms.
There have been a number of recent papers [ 1-6 ] dealing with the theoretical study of electron emission from multielectronic atoms by heavy charged particles; most of these concern the binary encounter model [ 13 ], the plane wave Born approximation [ 3,4 ] and the eikonal model [ 5,6 ]. Though it is stated that the classical model and the Born approximation are valid for high impact energies and over the intermediate range of ejection angles, these are often used in practice. Theoretical calculations of the double-differential cross section are difficult for even the simplest systems. Maybe this is the cause that the eikonal model is not widely used. For complicated systems, such as atoms with several electronic shells, there are further difficulties connected with the choice o f the correct atomic wave function. Recently the statistical model of an atom was used in this problem for the description of the initial state o f an atom [ 7,8 ]. Considerations were made using the binary encounter model. It is known that from the point of view o f comparison between theoretical and experimental results the binary encounter cross sections show a poorer agreement than the Born approximation. It may be argued that this is connected with the use o f the averaging procedure [ 1 ] in the classical model. The present work is a first attempt to connect the plane wave Born collision theory with the semiclassical WKB approximation for atoms. At least one can expect that an analytical solution of this problem will be obtained. According to recent considerations [ 3 ] the quantum-mechanical results for the double-differential cross section of/5-electron ejection are
dEtTnl# dee d-Qe -
Ac.~/2ks~ [~m~,(ks+ke-ki) [2 ki
iki_ks[4
dg2s,
(1)
where ~nt,(k) is the initial state wave function of an atomic electron in the m o m e n t u m representation,
k=ks+ke-ki, A=25/2Z2rn2m3/2e4/h 7. Here hki and hks denote the momenta o f the incident and scattered charged particles, hk is the m o m e n t u m of the atomic electron, hke is the m o m e n t u m of the electron after the collision, ¢~ is the energy o f the ejected electron, Zle is the charge o f the incident particle of which the mass equals ml and m is the electron mass. The quantities g2s and ~2~ are the solid angles for the scattered heavy charged particle and the ejected electron. Expression ( 1 ) corresponds to the electronic shell of which the principal, orbital and magnetic quantum numbers are n, I and/1, respectively. We are here going to use the semiclassical approximation as a limit of large quantum numbers. Therefore, according to the correspondence principle [ 9 ] the final state wave function of the electron is ~ f ( k ' ) = (27r) 3/26(k~ - k~). Straightforward application of the laws o f conservation o f energy and m o m e n t u m yields the following expression, Elsevier Science Publishers B.V.
469
Volume 173, number 6
PHYSICS LETTERSA
22 February 1993
2kq 1 dq dk dg2~= kZlqk~ Isin 0~sin O~sin q~l
(2 )
where O~ is the angle between ki and k e, 0 s the angle between k~ and k~, ~0~ is the spherical azimuthal angle of ks; q = k ~ - k i = k - k e . Expressing 0~ and ~0~in terms of 0~ one obtains the following relation, dea dG d(2~
_2Ae~/2~fdkf . J J aq kik~ 4kq Iq),au(k)l 2 ,,zu q4[ (q2 j ) (6_q2) ]1/2 ,
(3 )
for the total differential cross section. The factor 2 corresponding to the spin degeneracy is taken into account in eq. (3). The integration limits in (3) are defined by the condition 3 < q 2 < & where J=[-b-(b2-ac)l/2]/a, a=k2, + k 2 - 2 k i k e c o S O e ,
~=[-b+(h2-ac)'/2]/a, b= -- [ (ki.2 + k ~2) k ,2 ~sin20~+(ki-kecosOe)t'],
c = k ~2( k.2i - k ~.2) 2 sin2Oe +t'%~
l,=ki(k2
k 2 ) + k ~ ( k ?~_ k ~ ) c o s O ~
•
Using the usual representation 1
• ,zu{k) = ~ q,z(k) Yzu(O, {o) one obtains dZa d{ed-Qe-
A{le/2 f d k TIfq~/(k)I k2 ~(2l+1) (k2,
=
E,,/),
(4)
where " 2 2k~2 sin O~+k 2 - k ~ 2 + 2 k e ( k i - & ) cos O~ f ( k 2, E,z) = {4k2(k~ _ks) 2 sin20e + [k2__k2e + 2ke(k~-k~) c o s Oe]2} 3/2"
ki -- ks
m, ( ~ e - E,,;) ~/2ki
In the semiclassical approximation [10] Iqnll2=
k OEnt/On x [ k 2 - ( l + ½)Z/roe] }/2 dU(ro)/dro
(5)
where U is the potential energy of the atomic electron and ro the point of the stationary phase [10]: k2 = ~2m [E,l-- U(ro)] •
{6)
According to the semiclassical approximation summation (4) may be replaced by an integration. Taking into account that for given En~ the function f ( k 2, Enl) and ro do not depend on l one can obtain d2°" A~e]/2 ~ I d{edg2e- rtk2 droroe d k 2 f ( k 2 , h 2 k 2 / 2 m + U ( r o ) ) .
(7)
Here, the condition/max= kro corresponding to the semiclassical approximation was used. At l=,/min the centrifugal member in (5) must be equal to zero. Now, the integration limits in (7) are defined by the conditions O<~hZkZ/2m<~_U(ro),
k2>(k~-ks-kecosOe)
2,
(8)
where the condition En/< 0 is taken into account. In analogy with result (6) and ref. [ 10 ], calculating the semi470
Volume 173, number 6
PHYSICS LETTERSA
22 February 1993
classical m o m e n t u m wave function of the c o n t i n u u m one can obtain for given ro that
h2k2/2m=~e-U(ro) .
(9)
In this approximation the integration limits (8) are defined by the condition {1 - [ 1 - 2 cos 0e x ~ u ( r o )
+~+u(ro) ]~/2} 2 < rn2k2/m2k 2 <~u(ro).
(10)
The integration of (7) over all k's is carried out analytically. This yields r0
d 2a _ "~1,,1-72m3"~4 ! 4u(ro)_[e_2COSOe~]2 d ~ d£2e nh6k3~ 3/2 [ l _ 2 c o s O e ~ + E + u ( r o ) ] 3 / z r 2 d r ° '
(11)
where r ° is the solution of the equation 4 u ( r °) = [ e - 2 cos de ~
]z
(12)
and u( ro) = - 2m2U( ro) /h2mk 2 a n d e= 2m21ejh2mk 2 . The evaluation of the double-differential cross section is finished. Expressions ( 1 1 ) and (12) are the general solution of the problem. O f course, to be consistent and according to the semiclassical approximation and the plane wave Born approximation (1), one should have used the approximation ee= h2k 2/2m. But it is known that the energy dependence of the secondary electron spectrum is very sharp for large electron energies. Therefore, one can expect that approximation (9) improves the agreement of the theoretical and the experimental results. As a test, typical calculation results of the double-differential cross section in comparison with experimental data are shown in figs. 1, 2. Figure 1 corresponds to the range of very large energies ee and small ejection angles. Experimental data for silver (fig. l a ) [ 1 1 ] a n d copper (fig. l b ) [ 12] have been used in this work in order to test the above described model in the range of small angles de. A typical situation over an intermediate range
>
,
I 0
j
-19
~D
1 0 -2~ 1 0 -2o 0
10
¢.)
-26
z, %
%- \
\:
60
(,,0
b
1 0 -2,
1 0 -2~ 1 0 -22
1 0 -28
....
t~
%
% 1 0 -29 10 4
lO s
~e,
eV
Fig. 1. Double-differentialcross sectionsfor protons on silver (a) and copper (b) as a function of the electron energy. (a) T1=22 MeV, fl~= 5°; (b) T~= 10 MeV, tic= 5°. Experimental data from Bell et al. [ 1I, 12]. Solid lines are the results of the present work, T~ is the kinetic energyof initial particles.
1 0 -23 10 2
i
h
i
i
i iii
kb C J
10 3
ee,
eg
Fig. 2. Double-differentialcross sections for xenon as a function of the electron energy plotted for various proton kinetic energies T, and scatteringanglesOc. (a) T~= 300 keV, Oc= 20 ° ; (b) T, = 1 MeV, 0~=70°; (c) T~=2 MeV, 0~=90 °. Solid lines are the resuits of the present work. Experimentalpoints on xenon are from ref. [13]. 471
Volume 173, number 6
PHYSICS LETTERS A
22 February 1993
o f angles (0e~< 90 ° ) is d e m o n s t r a t e d in fig. 2. Here, the e x p e r i m e n t a l data lbr xenon [13] h a v e been used. T w o facts should be p o i n t e d out f r o m these c o m p a r i s o n s . Firstly, there is a g o o d a g r e e m e n t o f the c a l c u l a t e d results with e x p e r i m e n t for small and i n t e r m e d i a t e scattering angles ( 0 c < 9 0 ° ). As d e f i n i t i o n the T h o m a s - F e r m i m o d e l [14] was used to calculate the potential energy U ( r ) . It is possible that the use o f o t h e r a t o m i c p o t e n t i a l m o d e l s will give better results. Secondly, the c a l c u l a t i o n results are still in p o o r a g r e e m e n t for higher ejection angles. But this defect is not a c o n s e q u e n c e o f the a p p l i c a t i o n o f the semiclassical a p p r o x i m a t i o n . A possible e x p l a n a t i o n for this defect is the n u c l e a r a t t r a c t i o n o f the s e c o n d a r y electrons. So far it is not clear h o w to d e r i v e the semiclassical mom e n t u m c o n t i n u u m w a v e f u n c t i o n in the case o f an essential n u c l e a r attraction. It should be kept in m i n d that the semiclassical a p p r o x i m a t i o n is m o r e correct for large orbital q u a n t u m n u m b e r s , or large i m p a c t p a r a m e t e r s . In any case, b o t h t h e o r e t i c a l a n d e x p e r i m e n t a l d a t a are very small at large ejection angles in c o m p a r i s o n with those at small e j e c t i o n angles. Here, the c o m p a r i s o n a n d the discussion on the a g r e e m e n t h a v e necessarily been brief. T h i s is only a typical s i t u a t i o n in this field. A d e t a i l e d c o m p a r i s o n o f the present results with the e x p e r i m e n t a l d a t a and the doubledifferential cross section calculated using o t h e r m o d e l s [ 1,3,7,8 ] is a topic o f special discussion o u t s i d e the scope o f this article.
References [ l ] T.F.M. Bonsen and L. Vriens, Physica 47 (1970) 307. [2] M. Gryzinski, J. Phys. B 20 (1987) 4741. [3] L. Vriens, Physica 47 (1970) 267. [4] M.E. Rudd, Radiat. Res. 64 (1975) 153. [ 5 ] L. Wilets and S.J. Wallace, Phys. Rev. 169 ( 1968 ) 84. [6] P.D. Fainstein, V.H. Ponce and R.D. Rivarola, J. Phys. B 22 (1989) 1207. [7] S.A. Gerasimov and V.F. Volkov, Phys. Left. A 110 (1985) 276. [ 8 ] S.A. Gerasimov, Acta Phys. Hung. 60 ( 1986 ) 279. [9] J.D. Garcia, Phys. Rev. 159 (1967) 39. [ 10] S.A. Gerasimov, Czech. J. Phys. B 40 (1990) 425. [ 11 ] F. Bell, G. Trollmann, H. B6ckt and H.D. Betz, Nucl. lnstrum. Methods t 94 ( 1982 ) 423. [ 12 ] F. Bell, G. Trollmanm H. B6ckl and H.D. Betz, J. Phys. B 15 ( 1982 ) 1487. [ 13] L.H. Toburen, Phys. Rev. A 9 (1974) 2505. [ 14 ] P. Gombas, Die statische Theorie des Atoms und ihre Anwendungen (Springer, Berlin, 1949 ).
472
PHYSICS LETTERS A
Semiclassical model for 8-electron emission from multielectronic atoms S.A. Gerasimov Department of Nuclear Physics,Instituteof Physics, RostovState University,Rostov-on-Don344104, RussianFederation Received 2 September 1992; revised manuscript received 23 December 1992;accepted for publication 29 December 1992 Communicated by L.J. Sham
A new semiclassical simple model is presented to study electron emission by atoms during heavy charged particle collisions. This model is based on the use of the plane wave Born theory and the semiclassical WKB approximation for atoms.
There have been a number of recent papers [ 1-6 ] dealing with the theoretical study of electron emission from multielectronic atoms by heavy charged particles; most of these concern the binary encounter model [ 13 ], the plane wave Born approximation [ 3,4 ] and the eikonal model [ 5,6 ]. Though it is stated that the classical model and the Born approximation are valid for high impact energies and over the intermediate range of ejection angles, these are often used in practice. Theoretical calculations of the double-differential cross section are difficult for even the simplest systems. Maybe this is the cause that the eikonal model is not widely used. For complicated systems, such as atoms with several electronic shells, there are further difficulties connected with the choice o f the correct atomic wave function. Recently the statistical model of an atom was used in this problem for the description of the initial state o f an atom [ 7,8 ]. Considerations were made using the binary encounter model. It is known that from the point of view o f comparison between theoretical and experimental results the binary encounter cross sections show a poorer agreement than the Born approximation. It may be argued that this is connected with the use o f the averaging procedure [ 1 ] in the classical model. The present work is a first attempt to connect the plane wave Born collision theory with the semiclassical WKB approximation for atoms. At least one can expect that an analytical solution of this problem will be obtained. According to recent considerations [ 3 ] the quantum-mechanical results for the double-differential cross section of/5-electron ejection are
dEtTnl# dee d-Qe -
Ac.~/2ks~ [~m~,(ks+ke-ki) [2 ki
iki_ks[4
dg2s,
(1)
where ~nt,(k) is the initial state wave function of an atomic electron in the m o m e n t u m representation,
k=ks+ke-ki, A=25/2Z2rn2m3/2e4/h 7. Here hki and hks denote the momenta o f the incident and scattered charged particles, hk is the m o m e n t u m of the atomic electron, hke is the m o m e n t u m of the electron after the collision, ¢~ is the energy o f the ejected electron, Zle is the charge o f the incident particle of which the mass equals ml and m is the electron mass. The quantities g2s and ~2~ are the solid angles for the scattered heavy charged particle and the ejected electron. Expression ( 1 ) corresponds to the electronic shell of which the principal, orbital and magnetic quantum numbers are n, I and/1, respectively. We are here going to use the semiclassical approximation as a limit of large quantum numbers. Therefore, according to the correspondence principle [ 9 ] the final state wave function of the electron is ~ f ( k ' ) = (27r) 3/26(k~ - k~). Straightforward application of the laws o f conservation o f energy and m o m e n t u m yields the following expression, Elsevier Science Publishers B.V.
469
Volume 173, number 6
PHYSICS LETTERSA
22 February 1993
2kq 1 dq dk dg2~= kZlqk~ Isin 0~sin O~sin q~l
(2 )
where O~ is the angle between ki and k e, 0 s the angle between k~ and k~, ~0~ is the spherical azimuthal angle of ks; q = k ~ - k i = k - k e . Expressing 0~ and ~0~in terms of 0~ one obtains the following relation, dea dG d(2~
_2Ae~/2~fdkf . J J aq kik~ 4kq Iq),au(k)l 2 ,,zu q4[ (q2 j ) (6_q2) ]1/2 ,
(3 )
for the total differential cross section. The factor 2 corresponding to the spin degeneracy is taken into account in eq. (3). The integration limits in (3) are defined by the condition 3 < q 2 < & where J=[-b-(b2-ac)l/2]/a, a=k2, + k 2 - 2 k i k e c o S O e ,
~=[-b+(h2-ac)'/2]/a, b= -- [ (ki.2 + k ~2) k ,2 ~sin20~+(ki-kecosOe)t'],
c = k ~2( k.2i - k ~.2) 2 sin2Oe +t'%~
l,=ki(k2
k 2 ) + k ~ ( k ?~_ k ~ ) c o s O ~
•
Using the usual representation 1
• ,zu{k) = ~ q,z(k) Yzu(O, {o) one obtains dZa d{ed-Qe-
A{le/2 f d k TIfq~/(k)I k2 ~(2l+1) (k2,
=
E,,/),
(4)
where " 2 2k~2 sin O~+k 2 - k ~ 2 + 2 k e ( k i - & ) cos O~ f ( k 2, E,z) = {4k2(k~ _ks) 2 sin20e + [k2__k2e + 2ke(k~-k~) c o s Oe]2} 3/2"
ki -- ks
m, ( ~ e - E,,;) ~/2ki
In the semiclassical approximation [10] Iqnll2=
k OEnt/On x [ k 2 - ( l + ½)Z/roe] }/2 dU(ro)/dro
(5)
where U is the potential energy of the atomic electron and ro the point of the stationary phase [10]: k2 = ~2m [E,l-- U(ro)] •
{6)
According to the semiclassical approximation summation (4) may be replaced by an integration. Taking into account that for given En~ the function f ( k 2, Enl) and ro do not depend on l one can obtain d2°" A~e]/2 ~ I d{edg2e- rtk2 droroe d k 2 f ( k 2 , h 2 k 2 / 2 m + U ( r o ) ) .
(7)
Here, the condition/max= kro corresponding to the semiclassical approximation was used. At l=,/min the centrifugal member in (5) must be equal to zero. Now, the integration limits in (7) are defined by the conditions O<~hZkZ/2m<~_U(ro),
k2>(k~-ks-kecosOe)
2,
(8)
where the condition En/< 0 is taken into account. In analogy with result (6) and ref. [ 10 ], calculating the semi470
Volume 173, number 6
PHYSICS LETTERSA
22 February 1993
classical m o m e n t u m wave function of the c o n t i n u u m one can obtain for given ro that
h2k2/2m=~e-U(ro) .
(9)
In this approximation the integration limits (8) are defined by the condition {1 - [ 1 - 2 cos 0e x ~ u ( r o )
+~+u(ro) ]~/2} 2 < rn2k2/m2k 2 <~u(ro).
(10)
The integration of (7) over all k's is carried out analytically. This yields r0
d 2a _ "~1,,1-72m3"~4 ! 4u(ro)_[e_2COSOe~]2 d ~ d£2e nh6k3~ 3/2 [ l _ 2 c o s O e ~ + E + u ( r o ) ] 3 / z r 2 d r ° '
(11)
where r ° is the solution of the equation 4 u ( r °) = [ e - 2 cos de ~
]z
(12)
and u( ro) = - 2m2U( ro) /h2mk 2 a n d e= 2m21ejh2mk 2 . The evaluation of the double-differential cross section is finished. Expressions ( 1 1 ) and (12) are the general solution of the problem. O f course, to be consistent and according to the semiclassical approximation and the plane wave Born approximation (1), one should have used the approximation ee= h2k 2/2m. But it is known that the energy dependence of the secondary electron spectrum is very sharp for large electron energies. Therefore, one can expect that approximation (9) improves the agreement of the theoretical and the experimental results. As a test, typical calculation results of the double-differential cross section in comparison with experimental data are shown in figs. 1, 2. Figure 1 corresponds to the range of very large energies ee and small ejection angles. Experimental data for silver (fig. l a ) [ 1 1 ] a n d copper (fig. l b ) [ 12] have been used in this work in order to test the above described model in the range of small angles de. A typical situation over an intermediate range
>
,
I 0
j
-19
~D
1 0 -2~ 1 0 -2o 0
10
¢.)
-26
z, %
%- \
\:
60
(,,0
b
1 0 -2,
1 0 -2~ 1 0 -22
1 0 -28
....
t~
%
% 1 0 -29 10 4
lO s
~e,
eV
Fig. 1. Double-differentialcross sectionsfor protons on silver (a) and copper (b) as a function of the electron energy. (a) T1=22 MeV, fl~= 5°; (b) T~= 10 MeV, tic= 5°. Experimental data from Bell et al. [ 1I, 12]. Solid lines are the results of the present work, T~ is the kinetic energyof initial particles.
1 0 -23 10 2
i
h
i
i
i iii
kb C J
10 3
ee,
eg
Fig. 2. Double-differentialcross sections for xenon as a function of the electron energy plotted for various proton kinetic energies T, and scatteringanglesOc. (a) T~= 300 keV, Oc= 20 ° ; (b) T, = 1 MeV, 0~=70°; (c) T~=2 MeV, 0~=90 °. Solid lines are the resuits of the present work. Experimentalpoints on xenon are from ref. [13]. 471
Volume 173, number 6
PHYSICS LETTERS A
22 February 1993
o f angles (0e~< 90 ° ) is d e m o n s t r a t e d in fig. 2. Here, the e x p e r i m e n t a l data lbr xenon [13] h a v e been used. T w o facts should be p o i n t e d out f r o m these c o m p a r i s o n s . Firstly, there is a g o o d a g r e e m e n t o f the c a l c u l a t e d results with e x p e r i m e n t for small and i n t e r m e d i a t e scattering angles ( 0 c < 9 0 ° ). As d e f i n i t i o n the T h o m a s - F e r m i m o d e l [14] was used to calculate the potential energy U ( r ) . It is possible that the use o f o t h e r a t o m i c p o t e n t i a l m o d e l s will give better results. Secondly, the c a l c u l a t i o n results are still in p o o r a g r e e m e n t for higher ejection angles. But this defect is not a c o n s e q u e n c e o f the a p p l i c a t i o n o f the semiclassical a p p r o x i m a t i o n . A possible e x p l a n a t i o n for this defect is the n u c l e a r a t t r a c t i o n o f the s e c o n d a r y electrons. So far it is not clear h o w to d e r i v e the semiclassical mom e n t u m c o n t i n u u m w a v e f u n c t i o n in the case o f an essential n u c l e a r attraction. It should be kept in m i n d that the semiclassical a p p r o x i m a t i o n is m o r e correct for large orbital q u a n t u m n u m b e r s , or large i m p a c t p a r a m e t e r s . In any case, b o t h t h e o r e t i c a l a n d e x p e r i m e n t a l d a t a are very small at large ejection angles in c o m p a r i s o n with those at small e j e c t i o n angles. Here, the c o m p a r i s o n a n d the discussion on the a g r e e m e n t h a v e necessarily been brief. T h i s is only a typical s i t u a t i o n in this field. A d e t a i l e d c o m p a r i s o n o f the present results with the e x p e r i m e n t a l d a t a and the doubledifferential cross section calculated using o t h e r m o d e l s [ 1,3,7,8 ] is a topic o f special discussion o u t s i d e the scope o f this article.
References [ l ] T.F.M. Bonsen and L. Vriens, Physica 47 (1970) 307. [2] M. Gryzinski, J. Phys. B 20 (1987) 4741. [3] L. Vriens, Physica 47 (1970) 267. [4] M.E. Rudd, Radiat. Res. 64 (1975) 153. [ 5 ] L. Wilets and S.J. Wallace, Phys. Rev. 169 ( 1968 ) 84. [6] P.D. Fainstein, V.H. Ponce and R.D. Rivarola, J. Phys. B 22 (1989) 1207. [7] S.A. Gerasimov and V.F. Volkov, Phys. Left. A 110 (1985) 276. [ 8 ] S.A. Gerasimov, Acta Phys. Hung. 60 ( 1986 ) 279. [9] J.D. Garcia, Phys. Rev. 159 (1967) 39. [ 10] S.A. Gerasimov, Czech. J. Phys. B 40 (1990) 425. [ 11 ] F. Bell, G. Trollmann, H. B6ckt and H.D. Betz, Nucl. lnstrum. Methods t 94 ( 1982 ) 423. [ 12 ] F. Bell, G. Trollmanm H. B6ckl and H.D. Betz, J. Phys. B 15 ( 1982 ) 1487. [ 13] L.H. Toburen, Phys. Rev. A 9 (1974) 2505. [ 14 ] P. Gombas, Die statische Theorie des Atoms und ihre Anwendungen (Springer, Berlin, 1949 ).
472