ISSN 19907931, Russian Journal of Physical Chemistry B, 2009, Vol. 3, No. 5, pp. 737–742. © Pleiades Publishing, Ltd., 2009. Original Russian Text © I.Yu. Yurova, I.D. Borispol’skii, 2009, published in Khimicheskaya Fizika, 2009, Vol. 28, No. 10, pp. 28–34.
ELEMENTARY PHYSICOCHEMICAL PROCESSES
Theoretical Study of the Ionization of OrbitalPolarized pExcited Alkali Atoms by Electron Impact. 3D Representation of the Results I. Yu. Yurova and I. D. Borispol’skii Research Institute of Physics, St. Petersburg State University, Ul’yanovskaya ul. 1, Petrodvorets, 198904 Russia email: inna
[email protected];
[email protected] Received April 29, 2008
Abstract—The BornCoulomb approximation was used to study the electron impact ionization of pexcited Li and Na atoms with a given projection m of the orbital angular momentum on a specified axis. The wave function of the initial state of the atom electron was calculated using the effective charge approximation. The transformation of the initial wave function upon rotation of the coordinate system was considered. An ana lytical expression for the ionization amplitude was derived. Numerical results for the angular distributions of double differential cross sections of ionization were represented in the form of threedimensional images. DOI: 10.1134/S1990793109050066
Studies of the ionization of gases by electron impact are of considerable interest for the physics of the atmosphere and plasma [1–7]. Modern laser tech nique [5–7] makes it possible to investigate the ioniza tion of orbitalpolarized excited atoms, i.e., atoms in a state with the specified magnetic quantum number m. The authors of [4] revealed a difference in the angular distributions of triple differential cross sections of ion ization of excited sodium atoms at m = +1 and –1. This phenomenon is known as ionization dichroism. In the present work, we examined the possibility of observing dichroism in double differential cross sec tions of ionization and, in addition, considered the dependence of the cross section on the direction of the Z axis onto which the orbital angular momentum operator of the atom electrons is projected. This axis will be termed the quantization axis and denoted as O in the laboratory coordinate system; the same notation will be used for the unit vector in the O axis direction. Attempt to take into account the direction of the quantization axis were undertaken in [14, 15], but no numerically results were obtained, probably due to the complexity of the expressions derived therein. In the present work, the drawbacks of works [14, 15] are over come due to application of an analytical method to determining the quantities required to calculate the scattering cross section. As a theoretical method admitting analytical calcu lations, we used the modified first Born approximation with the Coulomb wave function for the continuous spectrum. This approximation is valid at energies of primary electrons E0 of 90–100 eV and higher and energies of secondary electrons of 20–25 eV and
higher [8]. The ionization cross section is given by (here and below, atomic units are used) 3 4kk ort (O) 2 dσ = 41 〈 k exp ( iqr ) nlm 〉 , dΩ 1 dΩ 2 de 2 k0 q
(1)
where q = k0 – k1 is the momentum imparted; k0 and k1 are the momenta of the primary electron in the ini tial and final states, respectively; k is the momentum of the secondary (ionized) electron; |nlm(O)〉 and |kort〉 are the wave functions of the initial bound state of the electron and of the final continuousspectrum state of the ionized electron [11]. To describe the initial state of a orbitalpolarized atom, the angular part of the wave function |nlm(O)〉 (O)
(spherical function Y lm ) was chosen dependent on the angles ϑO and ϕO defined in a coordinate system with the Z axis coinciding with the quantization axis O (this coordinate system will be termed the Osystem). Along with the Osystem, we introduced the ksystem of coordinates, with the Z axis directed along the k vector. The spherical angles and wave functions speci fied in the ksystem will be marked by the index “k.” The introduction of the ksystem was motivated by the circumstance that this system makes it possible to derive analytical expressions for the matrix elements 〈kort|exp(iqr)|nlm(O)〉 in formula (1) [9–17]. Using the transformation of the spherical functions for the rota tion of the coordinate system [18], we can write the
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YUROVA, BORISPOL’SKII
following relationship between the wave functions |nlm(O)〉 and |nlm(k)〉: (O)
|nlm 〉 =
∑D
l (k) m'm ( Ω k ) |nlm' 〉.
Using the explicit expressions for the Wigner func tions at l = 1 [18], we obtained the following expres sions for matrix elements (5):
(2)
〈k
m'
(O)
n1
exp ( iqr ) nl0 〉 = – 2 cos ( α ) sin ( β )F k1
(6a)
l D m'm (Ωk)
Here, is the Wigner function; Ωk defines the set of Euler angles, α, β, and γ, that relate by rotation the k and Osystems of coordinates, with the spheri cal angles ϑk and ϕk being transformed into the angles ϑO and ϕO. The wave function |kort〉 in formula (1) was pre sented as [9] (k)
ort
ort
(k)
|k 〉 = |k〉 – |nl0 〉 〈nl0 |k〉.
+
n1 cos ( β )F k0
(k)
+ [ sin (β) cos (α)A – cos ( β )B ] 〈 k n10 〉 , 〈k
ort
(O)
−1 〉 exp ( iqr ) nl + − iγ
= ± e+
nl sin β nl C +− F k1 + F k0 2
(6b)
– [ AC +− + sin ( β )B ] 〈 k n10〉 / 2 ],
(3)
where A = sinϑkqcosϑkq( G 11 – G 00 ), B = sin2ϑkq G 11 +
Here, |k〉 is the Coulomb wave function of an electron in the field of a positive unit charge [8]:
cos2ϑkq G 00 , C± = cos(β)cos(α) ± isin(α), and θkq is the angle between the vectors k and q. The Euler angle γ enters into formula (6b) only as a phase factor common to all summands, and therefore, knowledge of γ in calculating the ionization cross sec tion (1) is not required. Let us determine the values of the Euler angles α and β in expression (6). For this purpose, it is necessary to specify the directions of the axis in the coordinate systems used. Let the Zlab axis be directed along the vector k0 and let the Xlab axis lie in the Ok0 plane with negative projection of the vector O onto the Xlab axis. Let us define Osystem of coordi nates so that the ZO axis be parallel the vector O, the XO would lie in the XlabZlab plane, forming a sharp angle with the axis XO. Let the Xk axis lie in the plane passing through the vectors k and q in the k coordinate system, with the projection of the vector q onto the Xk axis being positive. This suggests that the Euler angle β is equal to the angle formed by the vectors k and O:
1 e π/2k Γ ⎛1 + i ⎞ ikr F ⎛– i , 1, – ikr – ikr⎞ (4) |k〉 = e , 3/2 ⎝ k⎠ 1 1 ⎝ k ⎠ ( 2π ) where 1F1 is the degenerate hypergeometric function, Γ is the gammafunction. Taking into account prop erty (2) and the property of independence of Coulomb wave function (4) of the angle ϕk (from which the equality 〈k|nlm(k)〉 = δ0m〈k|nl0(k)〉 follows), it is easy to see that the continuous spectrum wave function |kort〉 (formula (3)) is orthogonal to the wave function of the initial state |nlm(O)〉. Using transformation (2) and formula (3) makes it possible to present the matrix elements 〈kort|exp(iqr) nlm(O)〉 in expression (1) in the form of a linear combination of analytically calculated ele nl ments 〈k|exp(iqr)|nlm(k)〉, denoted thereafter as F km , and of elements 〈nlm(q)|exp(iqr)|nlm(q)〉, denoted by nl G m2 m2 : 〈k
ort
(O)
exp ( iqr ) nlm 〉 =
∑D
l m1 m ( Ω k )
nl
F km
m1
(k)
– 〈 k nl0 〉
∑
l
nl
nl
nl
cos β = kO/k.
(7)
For the cosine of the Euler angle α, we obtained the following expression: 2
∫ (5)
D m*2 0 ( Ω q )D m2 m1 ( Ω q )G m2 m2 . l
nl
nl
m2
Here, Ωq is the set of Euler angles that specify the rota tion from the k to the qsystem, with the qsystem of coordinates having the Z axis parallel to the imparted momentum vector. Note that, when deriving for mula (5), we used the relationship between wave func tions specified in the k and qsystems, which was determined by applying a transformation similar to nl that given by (2). As a result, elements G m2 m2 diagonal in the index m2 appeared in formula (5).
– kq ( kO/k ) cos α = qO , q sin θ kq sin θ kO
(8)
where θkq is the angle between the vectors k and q, θkO is the angle between the vectors k and O, which, according to (7), equals the Euler angle β. Note that, using formula cos θ qO = cos θ qk cos θ kO + sin θ qk sin θ kO cos ( ϕ q – ϕ O ), one can recast expression (8) as cosα = cos(ϕq – ϕO), where the azimuth angles ϕq and ϕO of the vectors q and O are defined in the coordinate system with the Z axis directed along the vector k. Note also that expres sion (8) does not specify the sign of sinα, which is required to calculate the quantities C± in formula (6b). Using a geometric construction, we found that the sign
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of sinα coincides with the sign of the projection of the O axis onto the Yk axis: sgn ( O [ kq ] ) = sgn { sin α }.
1 ⎛ 1 – 3α A 311 = ⎞ , 2δ ⎝ 4δ ⎠
(9)
Note that, in formula (5), knowledge of the angle αq is not required; in addition, a proper choice of the Xq axis of the qsystem of coordinates in the qk plane makes it possible to reduce the angle γq to zero, so that it suffices to know the value of only one Euler angle, βq = θkq. nl
nl
expression for the matrix element F km at n = 1, l = 0, and m = 0 was derived in [10]. The matrix elements 2l 3l F k0 and F k0 were obtained in [11] (in [11], the expres 2l ( k )
sion for F k0 contained an error, which was corrected in [12]). In the present work, we obtained expressions nl for the matrix elements F km at n = 2, 3, l = 1, and 21
m = ±1, 0; the expression for F k0 turned out to be identical with the analogous formula derived in [12]: 21 1 ( A J + A J + A J ), F k0 = C 200 0 201 1 202 2 8δ
2
(10)
− + iν ( A J + A J ), = C 211 1 212 2 2 8 2δ
Here, Z α = ef , n
ν = q sin ϑ kq ,
2 A 200 = – 2 , δ
A 201 = ν3 , δ
A 211 = 2 , δ
and – 1/2
5/2 ⎧
2π⎞ ⎫ C = α ⎨ πk 1 – exp ⎛ – ⎝ k⎠ ⎬ ⎩ ⎭
21 1 – 5t , G 00 = 2 4 (1 + t )
21 1 , G ±1, ±1 = 2 3 (1 + t )
1 ⎛ 3α – 2⎞ A 300 = , 2⎝ δ ⎠ δ
ν , A 202 = 1 – 4 16δ
2
ν ; A 212 = 1 – 2 4δ
2
4
6
2
4
6
α ν ν – ν ⎞ , A 303 = ⎛ – 1 + 2 + 4 6 4⎝ 4δ 16δ 64δ ⎠ RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
(13)
4
To perform numerical calculations of the ioniza tion cross sections by formulas (1), (5)–(13), it is nec essary to know the effective charge Zef for the wave function of the initial state. According to [23] 2
1 3n – l ( l + 1 ) Z ef = , 2 r nl
(12b)
α – αν ν , A 302 = 1 – + 9αν – 4δ 4δ 3 64δ 5 16δ 4
6
31 1 – 3t + 4t . G ±1, ±1 = 2 5 (1 + t )
2
2 1 1 α 9α A 301 = 2 + ν ⎛ – 2⎞ , ⎝ δ 8δ ⎠ δ 2
4
31 1 – 18t + 41t – 20t G 00 = , 2 6 (1 + t )
4
− 31 + iν ( A J + A J + A J ), F k ± 1 = C 311 1 312 2 313 3 2 3δ
exp { i arg ( Γ ( z ) ) },
where arg (Γ (z)) is the phase of the gammafunction at z = 1 + i/k. The expressions for Ji and derivation thereof are given in the Appendix (formulas (A1) and (A2)). The overlap integrals 〈k|n10(k)〉 can be obtained from formulas (10) and (12a) by setting q = 0 in them. Using the standard integrals for the matrix ele nl ments G mm , we obtained the following expressions dependent on the variable t = nq/2Zef:
(11)
31 2 ( A J + A J + A J + A J ), (12a) F k0 = C 300 0 301 1 302 2 303 3 4 3δ
2
ν λ = δ* + , 4δ
δ = [ α + iν – ik ]/2,
2
2
4
ν ⎞ ν – A 313 = α ⎛ 1 + . 8 ⎝ 2 8δ 2 32δ 4⎠
2
21
2
α + ν ⎛ 3α – 1⎞ , A 312 = 1 – ⎠ 4 4δ 16δ 2 ⎝ 2δ
nl
Let us calculate the matrix elements F km and G mm in formula (6). For this purpose, the radial part of the wave function |nlm(k)〉 was approximated by a hydro genlike wave function with effective charge Zef [10– 12] (definition of Zef is given below). An analytical
Fk ± 1
739
where n and l are, respectively, the principal quantum and angular momentum quantum numbers of the ini tial electron wave function and r nl is the mean dis tance between the atom electron and nucleus in the initial state, the socalled effective radius. The value of r nl was calculated using a wave function determined as a numerical solution to the Schrödinger equation with the effective potential [16]. Note that, despite the sim plicity of the hydrogenlike wave function used, the Vol. 3
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d3σ/dΩ1/dΩ2/de2, а.u.
12
m = −1
m=1
8
4
0
20
40
60
80
100
120 θ2, deg
Fig. 1. Triple differential cross sections of ionization, d3σ/dΩ1/dΩ2/de2 for a 3p excited Na atom (m = ±1) at ω = 90°, E0 = 150 eV, e2 = 20 eV, θ1 = 20°, ϕ1 = 90°, ϕ2 = 270°. Theory: the present work (solid lines) and [4] at m = 1 (dashed line) and m = –1 (chain dotted line); experiment: data from [4] for m = 1 (open circles) and m = –1 (closed circles).
obtained values of the ionization cross section are rather accurate. The accuracy of the calculations was tested by comparing the values of the total ion ization cross section determined using the hydro genlike wave functions with more accurate Har tree–Fock–Roothaan wave functions for oxygen, as an example [12]. We considered the ionization of a lithium atom from the 2p state and a sodium atom from the 3p state. For the sodium atom, we used the previously obtained value of the mean radius and the effective charge for the 3р state: r (Na(3p)) = 5.76a0, Zef(Na(3p)) = 2.17e (e is the charge of electron) [16]. By analogy with [16], we used the effective potential method to calcu late the mean radius and effective charge of the core in the 2p state for the lithium atom: r (Li(2p)) = 4.63a0, Zef(Li(2p)) = 1.08e. We compared our results on the tri ple differential cross section for the 3р ionization of the lithium atom at m = ±1 with those reported in [4], a work in which the distorted wave method was used to obtain numerical results for a complanar geometry of ionization at an angle ω between the quantization axis O and the vector k0 of 90° (Fig. 1) (we normalized the experimental data from [4] by the values obtained in the present work at θ2 = 57° and m = 1). Note that, in [4], the ionization cross section at m = 0 turned out to be zero at all values of the angle ϑ2, which is also pre dicted by formula (6a) (see above). Indeed, taking into account the values of the scattering angles (ϑ1 = 20°, ϕ1 = 90°, and ϕ2 = 270°) and the complanar geometry of the experiment, we obtained the following values of the Euler angles (7)–(9) entering into formula (6a):
α = 90° and β = 90°. As can be seen from Fig. 1, our results are in complete agreement with the experi mental data and are identical in form to the results of theoretical work [4]. Although the phenomenon of dichroism in the ion ization of orbital polarized atoms by electron impact has been observed only for the triple differential cross section [4], we examined manifestations of dichroism in the double differential cross section of ionization, taking into account the possibility of an experimental observation of it [4, 20] and the possibility of represen tation of the results in the form of threedimensional images and sections thereof. The results of calcula tions of the double differential cross sections of ioniza tion of a 2рexcited lithium atom and a 3рexcited sodium atom are displayed in Fig. 2. Planar sections of the threedimensional images clearly show manifesta tions of dichroism, especially at low energy of second ary electrons. As can be seen from formula (6b), ion ization dichroism is especially pronounced when the quantization axis O is directed perpendicularly to the direction of the initial momentum of primary elec trons k0 (ω = 90°); when the O axis is parallel to the vector k0 (ω = 0°), dichroism disappears completely. The above analysis led us to the following conclu sions. (1) The proposed model takes into account arbi trary orbital polarization of ionized atoms and the excitation of the atom in the initial state. For this pur pose, we introduced different coordinate systems and obtained explicit expressions for the Euler angles that specify the rotations that convert the coordinate sys tems into each other.
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741
(а)
20
20
0
0
−20
O 0 20
−20
0
20
−20
0
20
(b) 2 2
0
0 −2 0
−2
0
−2
2
0
2
(c) 20 20 0
0
−20
−20 20
0
20
−40
−20
0
(d)
−20
0
20
−2
0
2
40
6
4 4 2 2 0 0
−2
−2
0 2
0 4
−4
4
2
Fig. 2. On the left side: the angular distributions of d2σnlm(ϑ2, ϕ2)/dΩ2/de2 (10–19 cm2/Sr/eV) at m = 1. The ionization of (a, b) a 2р excited Li atom and (c, d) 3p excited Na atom at ω = 90°, E0 = 150 eV, e2 = 3 eV (a, c), and e2 = 20 eV (b, d); the vectors k0 and O in Figs. 2b–2d are directed as in Fig. 2a. On the right side: sections of the threedimensional images presented on the left side at m = 1 (solid lines) and –1 (dotted lines) in the plane perpendicular to the quantization axis O. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
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(2) The closed expression we used for the Coulomb wave function of the secondary electron (formula (4)) and hydrogenlike functions that approximate the radial parts of the wave functions of the initial state made it possible to obtain analytical expressions for the ionization amplitudes in the Born–Coulomb approximation, an approach that enormously simpli fies calculations of the ionization cross sections for this process (compare with the results of [14, 15]). (3) The theoretical method proposed in the present work makes it possible to perform numerical calcula tions of the ionization cross section within a reason ably short time even on a personal computer. It also offers the possibility of studying the angular distribu tion of secondary electrons in the double differential cross section of ionization with the help of 3D images. APPENDIX nl
The matrix elements F km = 〈k|exp(iqr)|nlm〉 were calculated analytically using the parabolic coordinates ξ, η, ϕ and the change of variables u = η cos ϕ, v = η sin ϕ. As a result, the following integrals were obtained: ∞
Jn =
n – λξ
∫ξ e
⎛ i 1, ikξ⎞ dξ. ⎠ k
1F1 ⎝ ,
(A.1)
0
Employing the relationship between 1F1 absorbed the Whittaker functions [22] and tables of Laplace integral transformation [21] yielded the following analytical expressions for integrals (A1): J n = Γ ( n + 1 )λ
–( n + 1 )
⎛ i n + 1; 1, ik ⎞ . k λ⎠
2F1 ⎝ ,
After applying the Euler transformation [22], 2F1 ( a,
b; c, z ) = ( 1 – z )
–a
⎛
2F1 ⎝ a,
z c – b; c; ⎞ ; z – 1⎠
the integrals Jn can be written as –n ⎛
j ⎞ Cn J n = J 0 n!λ ⎜ 1 + a ⎟, j j ⎝ ( ik – λ ) ⎠ j=1 n
∑
n > 0,
where j
a 1 = 1;
aj =
∏ { 1 – ( p – 1 )ik },
p=2
j > 1;
(A.2)
⎧ 2k ⎞ ⎫ exp { – i ln ( Φ ) }, J 0 = 1 exp ⎨ – 1 arctan ⎛ 2 2 ⎬ ⎝ λ 1+q –k ⎠⎭ ⎩ k j > 1; Φ = 1 – ik . λ REFERENCES 1. M. A. Coplan, J. H. Moore, and J. P. Doering, Rev. Mod. Phys. 66, 985 (1994). 2. R. Goruganthu, W. Wilson, and R. Bonham, Phys. Rev. A 35, 540 (1987). 3. A. Green and T. Savada, J. Atmosph. Terr. Phys 31, 1719 (1972). 4. A. Dorn, A. Elliot, J. Lower, E. Weigold, and J. Berak dar, Phys. Rev. Lett. 80, 257 (1998). 5. J. Nijland, J. D. Gouw, H. Dijkerman, and H. Nejde man, J. Phys. B 25, 2841 (1992). 6. C. Opal, E. Beaty, and W. Peterson, Atomic Data 4, 209 (1972). 7. M. E. Rudd, Nucl. Instrum. Methods Phys. Res. B 56, 162 (1991). 8. R. K. Peterkop, Theory of Atom Ionisation by Electron Collision (Zinatne, Riga, 1975) [in Russian]. 9. K. Omidvar, H. Kyle, and E. Sullivan, Phys. Rev. A 5, 1174 (1972). 10. H. Massey and C. Mohr, Proc. R. Soc. London, Ser. A 140, 613 (1933). 11. G. Peach, J. Phys. B 1, 1088 (1968). 12. I. Yu. Yurova and I. D. Borispol’skii, Vestn. SPbGU, Ser. 4, No. 3, 9 (2005). 13. P. Bartlett and A. Stelbovics, Phys. Rev. A 66, 012707 (2002). 14. D. Belkic, J. Phys. B 17, 3629 (1984). 15. K. Glemza and A. Kupliauskene, Lithuan. J. Phys. 45, 339 (2005). 16. I. Yurova, Phys. Rev. A 65, 032726 (2002). 17. P. Bartlett and A. Stelbovics, Atom. Data 86, 235 (2004). 18. D. A. Varshalovich, A. N. Moskalev, and V. K. Kher sonskii, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975; World Sci., Singapore, 1988). 19. I. V. Hertel and W. Stoll, Adv. At. Mol. Phys. 13, 113 (1977). 20. M. Abramovich and I. Stegun, Handbook on Special Functions (Nauka, Moscow, 1979) [in Russian]. 21. G. Bateman and A. Erdelyi, Tables of Integral Trans forms (McGrawHill, New York, 1954; Nauka, Mos cow, 1969), Vol. 1. 22. G. Bateman and A. Erdelyi, Higher Transcendent Func tions (McGrawHill, New York, 1953; Nauka, Mos cow, 1965), Vol. 1. 23. A. S. Davydov, Quantum Mechanics (Fizmatlit, Mos cow, 1963; Pergamon Press, Oxford, 1976).
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