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Electron excitation cross sections of Li, Na and K
Volume 64A, number 4
PHYSICS LETTERS
ELECTRON
EXCITATION
9 January 1978
C R O S S S E C T I O N S O F Li, Na A N D K D.N. ROY
Department of Physics, Patna University,Patna 800005, India Received 4 August 1977 The Vainshtein approximation with exchange via the Ochkur-Rudge method and Ochkur's modified first Born approximation are used to calculate electron excitation cross sections of Li, Na and K. The results are compared with experiments and the frozen core Glauber cross sections.
In a review on Glauber approximation Gerjuoy and Thomas [1 ] have emphasized the need of comparing Glauber cross sections with other competing approximations preferably using the same wavefunctions. Waiters [2] has reported the frozen core Glauber cross sections for the resonance transitions in Li, Na and K and has compared his results with those of the Vainshtein approximation (VPS) calculated by Vainshtein et al. [3]. But the wavefunctions used in ref. [3] are different from those used in ref. [2]. Further, Coleman [4] has pointed out that a wrong expression has been used for the exchange amplitude in the VPS approximation. He has suggested that exchange could be included more simply via the O c h k u r - R u d g e method. We have used the VPS approximation with exchange via the Ochkur-Rudge method and the modified first Born approximation [5] to calculate these cross sections, using for Na and K the same wavefunctions as in ref. [2] and for Li a simple Slater wavefunction. Our VPS results are in better agreement with experiments than those of ref. [3] and are not inferior to Glauber cross sections as reported in ref. [2]. In the "post form" of the VPS approximation [4], the integral cross section for direct excitation (in atomic units)is given by
z = [(q2 + Ae)/(3q2 +
Ae)l ,
u = - (k 0 +x/U) -1 ,
(2)
t~i and t~f being the initial and final state wavefunctions. Here k 2 is the incident energy in rydbergs, AE is the threshold of excitation, q is the m o m e n t u m transfer, U the ionization energy of the initial state and 2F1 is the usual hypergeometric function. With exchange included via the O c h k u r - R u d g e method, the total integral cross section becomes qmax Ot = (8/k2) qmin f
q4
q2 cos(20- ]
(k2+ g l ) 2
( g l + k 2) _j
+ I1
(3)
X I¢(q)12[h(z, v)12q-3dq, with = tan-1 x/-U/~02 - AE),
(4)
where U 1 is the ionization energy of the final state. In the modified first Born approximation (MFBA), which takes into account the acceleration of the electron by the nucleus, the integral cross section [5] is given by t
q max a d = (8/k 2)
f
kb(q)12lh(z, v)[2q-3dq,
BM
(1)
kl k0, oB °(E)-kokl (E+2U)'
(5)
qmin where
with
k 0, = ( k 2 + 2 U ) 1/2;
O(q) =f4q(r) e~'" ~i(r) dr, h(z,
u) = (nv/sin hnu)2F 1 [ - i v , iv, 1, z],
k 1' = ( k 2 + 2 U ) 1/2.
(6)
k 0 and k 1 are the momenta of the incident and scattered electron and E the energy of the incident elec-
373
Volume 64A, number 4
PHYSICS LETTERS
9 January 1978
140
I
120
!
lO0
8oi
~o 6 o 3 4o
20
0
0-1
0"2
O'5
O.5 1-0 2,0 ENERGY III F~r.
5.o
~,.O 70
10
Fig. 1. Total excitation cross section for the process e-+ Li(2s) ~ e- + Li(2p). Curve 1: FBA; curve 2: VPS with exchange; curve 3: MFBA; all our calculations. Curve 4: frozen core Glauber approximation by Walters [2]; curve 5: normalized direct excitation cross section measured by Leep and Gallagher [9]. t r o n . o B is t h e first B o r n cross section. T h e cross s e c t i o n s for Li ( 2 s - 2 p ) , N a ( 3 s - 3 p ) a n d K ( 4 s - 4 p ) have b e e n c a l c u l a t e d using eqs. ( 3 ) a n d (5). The simple Slater w a v e f u n c t i o n used for Li [6] gives
a l m o s t t h e same first B o r n cross sections as t h e a c c u r a t e H F w a v e f u n c t i o n s o f H e r m a n n a n d S k i l l m a n n [7]. T h e results are graphically c o m p a r e d w i t h t h o s e o f t h e f r o z e n core G l a u b e r cross s e c t i o n o f ref. [2] a n d t h e r e c e n t ex-
120
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8o
"o 4o
2o
o
i
o.1
0.2
0.3
.
=
o.5
.
.
.
.
.
.
.
.
1.o
ENERGY
.
2:0
.
30
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5.o
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Fig. 2. Total excitation cross section for the process e- + Na(3s) ~ e- + Na(3p). Curves 1, 2, 3 and 4: the same as in fig. 1. Curve 5: normalized direct cross section measured by Enemark and Gallagher [8]. 374
Volume 64A, number 4
PHYSICS LETTERS
9 January 1978
'2O0
180
160
140
120
8O 4 60 40
o o.1
o.2
o.5
o.5 ENERGY
1.o
2.0
3-0
5.0
7-0
lo
IN Rr.
Fig. 3. Total excitation cross section for the process e- + K(4s) ~ e- + K(4p). Curves 1, 2, 3 and 4: the same as in fig. 1. perimental results. For Li and Na reliable experimental results have recently been reported [8, 9], which are shown in our figs. 1 and 2. But for K experimental results available are only thbse shown in ref. [2], hence these are not included in our fig. 3. The first Born cross sections are also shown in the figures. These are in reasonable agreement with other FBA calculations [7], providing a check on the accuracy of our calculations. For lithium (fig. 1) MFBA cross sections are in better agreement with experimental results over almost the entire energy range. Though above 1 rydberg, Glauber cross sections almost coincide with the experimental results, at lower energies these are considerably smaller. The VPS cross sections are consistently smaller (about 20%) than experimental results except at very low energies. For sodium (fig. 2) the VPS cross sections are in excellent agreement with experimental results down to 0.5 rydberg. Above 1 Ry the Glauber cross sections
are about 10% higher than VPS (and experiment) but close to MFBA cross sections. At lower energies Glauber cross sections are too low and MFBA too high. For potassium also the relative values of the three cross sections are nearly the same as those for Na. The inclusion of exchange via O c h k u r - R u d g e improves the VPS approximation at energies below the peak value, above which there is no significant change. The VPS and MFBA cross sections are no more difficult to evaluate than the FBA. The cross sections for several other transitions of Li, Na and K as well as of Rb have been calculated in these approximations and will be reported elsewhere. The author is extremely grateful to Professor L.S. Singh for suggesting the problem and taking keen inter. est in it. The work was supported by a research grant from the U.G.C., New Delhi.
375
Volume 64A, number 4
PHYSICS LETTERS
References [1] E. Gerjuoy and B.K. Thomas, Rept. Progr. Phys. 37 (1974). [2] H.R.J. Waiters, J. Phys. B6 (1973) 1003. [3] L. Vainshtein, V. Opykhtyn and L. Presnyakov, Soviet Phys. JETP 20 (1965) 1542. [4] J.P. Coleman, Case studies in atomic collision physics, Vol. 1, eds. E.W. McDaniel and M.R.C. McDowell (NorthHolland, Amsterdam, 1969).
376
9 January 1978
[5] V.I. Ochkur, Proc. 7th Intern. Conf. on Physics of electronic and atomic collisions, Amsterdam (1971) p. 755. [6] P. Simsic and W. Williamson Jr., J. Chem. Phys. 37 (1972) 4617. [7] T.J. Greene and W. Williamson Jr., Atomic Data Nucl. Data Tables 14 (1974) 161. [8] D. Leep and A. Gallagher, Phys. Rev. A10 (1974) 1082. [9] E.A. Enemark and A. Gallagher, Phys. Rev. A6 (1972) 192.
PHYSICS LETTERS
ELECTRON
EXCITATION
9 January 1978
C R O S S S E C T I O N S O F Li, Na A N D K D.N. ROY
Department of Physics, Patna University,Patna 800005, India Received 4 August 1977 The Vainshtein approximation with exchange via the Ochkur-Rudge method and Ochkur's modified first Born approximation are used to calculate electron excitation cross sections of Li, Na and K. The results are compared with experiments and the frozen core Glauber cross sections.
In a review on Glauber approximation Gerjuoy and Thomas [1 ] have emphasized the need of comparing Glauber cross sections with other competing approximations preferably using the same wavefunctions. Waiters [2] has reported the frozen core Glauber cross sections for the resonance transitions in Li, Na and K and has compared his results with those of the Vainshtein approximation (VPS) calculated by Vainshtein et al. [3]. But the wavefunctions used in ref. [3] are different from those used in ref. [2]. Further, Coleman [4] has pointed out that a wrong expression has been used for the exchange amplitude in the VPS approximation. He has suggested that exchange could be included more simply via the O c h k u r - R u d g e method. We have used the VPS approximation with exchange via the Ochkur-Rudge method and the modified first Born approximation [5] to calculate these cross sections, using for Na and K the same wavefunctions as in ref. [2] and for Li a simple Slater wavefunction. Our VPS results are in better agreement with experiments than those of ref. [3] and are not inferior to Glauber cross sections as reported in ref. [2]. In the "post form" of the VPS approximation [4], the integral cross section for direct excitation (in atomic units)is given by
z = [(q2 + Ae)/(3q2 +
Ae)l ,
u = - (k 0 +x/U) -1 ,
(2)
t~i and t~f being the initial and final state wavefunctions. Here k 2 is the incident energy in rydbergs, AE is the threshold of excitation, q is the m o m e n t u m transfer, U the ionization energy of the initial state and 2F1 is the usual hypergeometric function. With exchange included via the O c h k u r - R u d g e method, the total integral cross section becomes qmax Ot = (8/k2) qmin f
q4
q2 cos(20- ]
(k2+ g l ) 2
( g l + k 2) _j
+ I1
(3)
X I¢(q)12[h(z, v)12q-3dq, with = tan-1 x/-U/~02 - AE),
(4)
where U 1 is the ionization energy of the final state. In the modified first Born approximation (MFBA), which takes into account the acceleration of the electron by the nucleus, the integral cross section [5] is given by t
q max a d = (8/k 2)
f
kb(q)12lh(z, v)[2q-3dq,
BM
(1)
kl k0, oB °(E)-kokl (E+2U)'
(5)
qmin where
with
k 0, = ( k 2 + 2 U ) 1/2;
O(q) =f4q(r) e~'" ~i(r) dr, h(z,
u) = (nv/sin hnu)2F 1 [ - i v , iv, 1, z],
k 1' = ( k 2 + 2 U ) 1/2.
(6)
k 0 and k 1 are the momenta of the incident and scattered electron and E the energy of the incident elec-
373
Volume 64A, number 4
PHYSICS LETTERS
9 January 1978
140
I
120
!
lO0
8oi
~o 6 o 3 4o
20
0
0-1
0"2
O'5
O.5 1-0 2,0 ENERGY III F~r.
5.o
~,.O 70
10
Fig. 1. Total excitation cross section for the process e-+ Li(2s) ~ e- + Li(2p). Curve 1: FBA; curve 2: VPS with exchange; curve 3: MFBA; all our calculations. Curve 4: frozen core Glauber approximation by Walters [2]; curve 5: normalized direct excitation cross section measured by Leep and Gallagher [9]. t r o n . o B is t h e first B o r n cross section. T h e cross s e c t i o n s for Li ( 2 s - 2 p ) , N a ( 3 s - 3 p ) a n d K ( 4 s - 4 p ) have b e e n c a l c u l a t e d using eqs. ( 3 ) a n d (5). The simple Slater w a v e f u n c t i o n used for Li [6] gives
a l m o s t t h e same first B o r n cross sections as t h e a c c u r a t e H F w a v e f u n c t i o n s o f H e r m a n n a n d S k i l l m a n n [7]. T h e results are graphically c o m p a r e d w i t h t h o s e o f t h e f r o z e n core G l a u b e r cross s e c t i o n o f ref. [2] a n d t h e r e c e n t ex-
120
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8o
"o 4o
2o
o
i
o.1
0.2
0.3
.
=
o.5
.
.
.
.
.
.
.
.
1.o
ENERGY
.
2:0
.
30
.
5.o
|
7o
,
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IN Rr.
Fig. 2. Total excitation cross section for the process e- + Na(3s) ~ e- + Na(3p). Curves 1, 2, 3 and 4: the same as in fig. 1. Curve 5: normalized direct cross section measured by Enemark and Gallagher [8]. 374
Volume 64A, number 4
PHYSICS LETTERS
9 January 1978
'2O0
180
160
140
120
8O 4 60 40
o o.1
o.2
o.5
o.5 ENERGY
1.o
2.0
3-0
5.0
7-0
lo
IN Rr.
Fig. 3. Total excitation cross section for the process e- + K(4s) ~ e- + K(4p). Curves 1, 2, 3 and 4: the same as in fig. 1. perimental results. For Li and Na reliable experimental results have recently been reported [8, 9], which are shown in our figs. 1 and 2. But for K experimental results available are only thbse shown in ref. [2], hence these are not included in our fig. 3. The first Born cross sections are also shown in the figures. These are in reasonable agreement with other FBA calculations [7], providing a check on the accuracy of our calculations. For lithium (fig. 1) MFBA cross sections are in better agreement with experimental results over almost the entire energy range. Though above 1 rydberg, Glauber cross sections almost coincide with the experimental results, at lower energies these are considerably smaller. The VPS cross sections are consistently smaller (about 20%) than experimental results except at very low energies. For sodium (fig. 2) the VPS cross sections are in excellent agreement with experimental results down to 0.5 rydberg. Above 1 Ry the Glauber cross sections
are about 10% higher than VPS (and experiment) but close to MFBA cross sections. At lower energies Glauber cross sections are too low and MFBA too high. For potassium also the relative values of the three cross sections are nearly the same as those for Na. The inclusion of exchange via O c h k u r - R u d g e improves the VPS approximation at energies below the peak value, above which there is no significant change. The VPS and MFBA cross sections are no more difficult to evaluate than the FBA. The cross sections for several other transitions of Li, Na and K as well as of Rb have been calculated in these approximations and will be reported elsewhere. The author is extremely grateful to Professor L.S. Singh for suggesting the problem and taking keen inter. est in it. The work was supported by a research grant from the U.G.C., New Delhi.
375
Volume 64A, number 4
PHYSICS LETTERS
References [1] E. Gerjuoy and B.K. Thomas, Rept. Progr. Phys. 37 (1974). [2] H.R.J. Waiters, J. Phys. B6 (1973) 1003. [3] L. Vainshtein, V. Opykhtyn and L. Presnyakov, Soviet Phys. JETP 20 (1965) 1542. [4] J.P. Coleman, Case studies in atomic collision physics, Vol. 1, eds. E.W. McDaniel and M.R.C. McDowell (NorthHolland, Amsterdam, 1969).
376
9 January 1978
[5] V.I. Ochkur, Proc. 7th Intern. Conf. on Physics of electronic and atomic collisions, Amsterdam (1971) p. 755. [6] P. Simsic and W. Williamson Jr., J. Chem. Phys. 37 (1972) 4617. [7] T.J. Greene and W. Williamson Jr., Atomic Data Nucl. Data Tables 14 (1974) 161. [8] D. Leep and A. Gallagher, Phys. Rev. A10 (1974) 1082. [9] E.A. Enemark and A. Gallagher, Phys. Rev. A6 (1972) 192.