Ionization of K-shell electrons by electron impact

Physics Letters A 372 (2008) 4451–4461

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Physics Letters A www.elsevier.com/locate/pla

Ionization of K-shell electrons by electron impact A.I. Mikhailov a,b , A.V. Nefiodov a,c,∗ , G. Plunien c a b c

Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, D-01187 Dresden, Germany Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, Germany

a r t i c l e

i n f o

Article history: Received 7 March 2008 Accepted 13 March 2008 Available online 9 April 2008 Communicated by V.M. Agranovich PACS: 34.10.+x 34.80.-i 34.80.Dp

a b s t r a c t We have investigated the universal scaling behavior for cross sections of the single K-shell ionization by electron impact. The calculations are performed within the framework of non-relativistic perturbation theory, taking into account the one-photon exchange diagrams. Special emphasis is laid on the nearthreshold energy domain. The results obtained are applicable for wide family of atomic targets with moderate values of the nuclear charge Z . © 2008 Elsevier B.V. All rights reserved.

1. The single ionization of inner-shell electrons by the electron impact is one of the fundamental processes, which still attracts considerable interest in collision theory, experimental studies, and various applications [1–7]. To predict the ionization cross sections, one usually employs either sophisticated numerical approaches or empirical and semi-empirical formulas. Significant efforts have been also devoted to investigations of the scaling behavior of ionization cross sections with respect to the incident electron energy [8–14]. However, a consistent theoretical consideration of the problem, which would allow one to deduce the universal laws accurately describing the ionization processes for different atomic targets within the wide energy range from the ionization threshold to asymptotic high energies, appears to be still absent in the literature. In this Letter, we deduce the universal scaling behavior for cross sections of the single K-shell ionization of hydrogen-like multicharged ions with moderate nuclear charge numbers Z by electron impact. The energy domains are considered near the ionization threshold and far beyond it. The study is performed to leading order of non-relativistic perturbation theory with respect to the electron–electron interaction. The nucleus of an ion is treated as an external source of the Coulomb field. Accordingly, the Coulomb functions are employed as electron wave functions in a zeroth approximation (Furry picture). The parameter α Z , where α is the fine-structure constant, is supposed to be sufficiently small (α Z  1), although we assume that Z  1. The relativistic units are used (h¯ = 1, c = 1).

*

Corresponding author at: Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, Germany. E-mail address: nefi[email protected] (A.V. Nefiodov). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.03.062

The universal scaling we obtained turns out to be also applicable for a wide family of atomic targets with the closed K shell. 2. Let us consider first the inelastic high-energy scattering of an electron on hydrogen-like ion in the ground state, which results in ionization of a K-shell bound electron. We shall derive formulas for the differential and total cross sections of the process. An incident electron can be characterized by the energy E = p 2 /(2m) and the momentum p at infinitely large distances from the nucleus. We focus on asymptotic non-relativistic energies E within the range I  E  m, where I = η2 /(2m) is the threshold energy for single ionization from the K shell, η = mα Z is the average momentum of the bound electron, and m is the electron mass. The process under consideration is described by the Feynman diagrams depicted in Fig. 1. In the final continuum state, the electron wave functions are denoted as ψ p 1 and ψ p 2 . In the case of ionization by high-energy particle impact, the main contribution to the total cross section arises from the edge domains of the electron energy spectrum, where the energy of one outgoing electron is much larger than that of another electron. More specifically, it turns out that either E 1 ∼ E and E 2 ∼ I or E 1 ∼ I and E 2 ∼ E. In the following, we shall label the fast and slow electrons by

Fig. 1. Feynman diagrams for ionization of a K-shell electron by an electron impact. Solid lines denote electrons in the Coulomb field of the nucleus, while dashed line denotes the electron–electron Coulomb interaction.

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the indices 1 and 2, respectively. The asymptotic momenta of outgoing electrons are estimated as p 1 ∼ p  η , p 2 ∼ η , while the single edge domain is characterized by p 1  p 2 . Accordingly, one needs to take into account only the Feynman diagram depicted in Fig. 1(a). The contribution of the exchange diagram turns out to be of about a factor (η/ p )2 smaller and, therefore, can be neglected. The spin functions are suppressed throughout the consideration in this paragraph. Then the amplitude of the process reads



A=

df

df1

df2

(2π )3 (2π )3 (2π )3 × ψ p1 | f 1  f 1 + f |ψ p  D ( f )ψ p 2 | f 2  f 2 − f |ψ1s ,

(1)

2

where D ( f ) = 4πα / f is the photon propagator. Since p ∼ p 1  η , the wave functions of both incident and scattered high-energy electrons can be approximated by plane waves (the first Born approximation). Integrating over the intermediate momenta in Eq. (1), one receives

  ∂ − ψ p 2 | V i λ |q|λ=η , ∂λ q2 1

A = 4πα N 1s  f  | V iλ | f  =

4π 

(f − f

+λ

)2

, 2

(2)

2 where N 1s = η3 /π and q = p − p 1 is the momentum transfer. After taking the derivative with respect to λ, one should set λ = η . In Eq. (2), the matrix element can be cast into the following form [15]

ψ p 2 | V i η |q = N p 2 K p2 (q),  i ξ2 4π a K p2 (q) = , a

b

2π ξ2

N 2p 2 =

1 − exp(−2π ξ2 )

(14)

χ = |b|2 = q2 + η2 − p 2

(15)



2

2

b = q − ( p 2 + i η) ,

(8)

The differential cross section for ionization of a K-shell electron is related to the amplitude (8) as follows dσK+ =

2π

υ

|A|2 dΦδ ( E 1 + E 2 − E + I ),

d p1

(2π

)3

d p2

(2π )3

(9) (10)

.

Here υ = p /m is the absolute magnitude of the velocity of the incident electron. Eq. (9) defines the distributions over energy and ejection angles of the fast and slow electrons. The elements of phase volumes for the fast and slow electrons ejected into the solid angles dΩ1 and dΩ2 , respectively, can be written as d p 1 = mp 1 dE 1 dΩ1 = π

m p

dE 1 dq2 ,

d p 2 = mp 2 dE 2 dΩ2 .

(11) (12)

Integrating Eq. (9) over the energy E 1 and the angles of ejection of slow electrons yield dσK+ =

πα 2m2 η6 e 2γ ξ2 dE 2 u dq2 , 3 [1 − exp(−2π ξ2 )] E χ 3 q2

29

(16)

dσK+ dε2

σ0

=

Z4

F (ε , ε2 ) =

F (ε , ε2 ), 28 3ε

(17)

(13)

x2

sinh−1 (π ξ2 )

(3x + ) xw 3

e 2ξ2 arctan φ dx,

(18)

x1

where

ν = 1 − ε2 , (x + ν ) φ= √ , 2 ε2

(7)

 24 πη  2 q − (1 + i ξ2 )( p 2 · q) K p 2 (q). q2 ab

p 22 .

(4)

and ξ2 = η/ p 2 . The dimensionless parameter ξ2−1 is merely the momentum p 2 of slow electron, which is calibrated in units of the characteristic momentum η . Inserting Eqs. (3)–(7) into Eq. (2) and taking the derivative yield

A = α N 1s N p2

2

+ 4η2 p 22 ,

Note that the distribution over the ejection angles of fast electrons is reduced to the dependence on the square of the momentum transfer q2 . To obtain the energy distribution, the expression (13) should be integrated over q2 within the range from q2min = ( p − p 1 )2 to q2max = ( p + p 1 )2 . This integration can be performed only numerically. In the following, it is convenient to express all momenta and energies in units of the averaged momentum η = mα Z and the Coulomb ionization potential I = η2 /(2m), respectively. Let us introduce dimensionless quantities, such as x = q2 /η2 , ε = E / I , and εi = E i / I , (i = 1, 2). Energy conservation now implies ε − 1 = ε1 + ε2 . Then one obtains

(6)

2

 2 2

u = 3q + η +

x1 =

a = ( p 2 − q)2 + η2 ,



w = (x + ν )2 + 4ε2 ,

where

dΦ =



γ = arg(b) = arg q2 + η2 − p 22 − i2η p 2 ,

(3)

(5)

,

where

x2 =

q2min

η

2

q2max

η

2

= =

(19)

 = 1 + ε2 , 1

ξ2 = √ 2 x2

ε2

(20)

,

(22)

,

√

(21)

√

2

ε+ ε− .

(23)

In Eq. (17), σ0 = π a20 = 87.974 Mb, where a0 = 1/(mα ) is the Bohr radius. The dimensionless function F (ε , ε2 ) describes the universal energy distribution, which does not depend on the particular value of Z . In fact, Eq. (18) coincides with the function obtained by H. Bethe for the case of the hydrogen atom [16–18]. Although the use of the Born approximation should be legitimate only within the asymptotic non-relativistic range 1  ε  2(α Z )−2 , we have also performed the calculations near the ionization threshold, in order to compare them with the exact results, which will be obtained in the next paragraph. In Fig. 2, the energy distribution (18) is presented with respect to the energy of one of the outgoing electrons, without distinguishing between fast and slow particles. In each event of the ionization process, there appear two outgoing electrons, a fast and a slow one. Due to the energy-conservation law, the excess energy ε1 + ε2 is fixed by the energy ε of the incident electron, taking into account the reduction by the binding energy of the K-shell electron. Since the number of slow electrons is equal to the number of fast electrons, the energy distributions are symmetrical relative to the center point ε1 = ε2 of the energy interval. As seen from Fig. 2, already for the electron energies ε  4, the excess energy ε − 1 becomes distributed greatly non-uniformly among the outgoing electrons. Accordingly, the main contribution to the total cross section arises from the edge domains of the energy spectrum characterized by the emission of electrons with extremely different energy sharing. The total cross section for the electron-impact ionization of a K-shell electron can be written as

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Fig. 2. Evolution of the energy distributions for different values of the dimensionless energy ε of the incident electron. The energy ε  is either ε1 or ε2 , taking into account that one does not distinguish between fast and slow electrons. The center point ε  = (ε − 1)/2 corresponds to the equal-energy sharing (ε1 = ε2 ). Dotted line, calculation within the Born approximation according to Eq. (18); solid line, exact calculation according to Eq. (53).

σK+ =

σ0 Z4

Q (ε ),

(24)

(ε −1)/2

Q (ε ) =

• Due to the crossing symmetry, the Feynman diagram depicted

F (ε , ε2 ) dε2 ,

(25)

0

where F (ε , ε2 ) is given by Eq. (18). The quantity Q (ε ) is the universal scaling function of the dimensionless energy of an incident electron, which holds true within the asymptotic non-relativistic range 1  ε  2(α Z )−2 . In Table 1, numerical values for the function Q (ε ) are compiled. For energies ε  10, the calculations are performed according to Eqs. (18) and (25). The numerical data are in agreement with those given in the book [18]. In Fig. 3, we present also the near-threshold behavior of the function (25), in order to compare it with the exact calculations. Assuming the Bethe asymptotic expansion [1–3], the quantity Q (ε ) can be cast into the following form



Let us make the following comments:



Q (ε ) = ε −1 C 0 ln ε + C 1 + C 2 ε −1 .

(26)

For energies ε  10, the function (25) is accurately approximated by the formula (26), if the coefficients are equal to C 0 = 1.126, C 1 = 5.067, and C 2 = −11.19.

in Fig. 1(a) also describes the ionization of an K-shell electron by the positron impact. Since the interacting particles are not identical, the exchange effect is absent. To receive the amplitude for this process, one needs to make the following substitutions: p  − p 1 , which do not alter Eqs. (17)–(23). The total cross section is again given by the formula (24). However, the function Q (ε ) is now obtained by the integration of F (ε , ε2 ) over the energy range 0  ε2  ε − 1. This explains the difference in behavior of the curves Q (ε ), which describe the ionization by electron and positron impacts at low energies (see Fig. 4). For the positron energies 1  ε  2(α Z )−2 , the integral over the variable ε2 is saturated at the lower integration limit. Accordingly, the total cross sections for the K-shell ionization by the electron and positron impacts exhibit the same asymptotic behavior [4]. • What are modifications of the formulas obtained, if a K-shell bound electron is ionized by a fast charged projectile with mass M being different from the electron mass m? In this case, the exchange diagram is absent. The energy-conservation

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Table 1 For energies within the range 1 < ε < 10, the universal functions Q (ε ) and Z 4 σK+ (in Mb) are calculated according to Eqs. (53)–(55). For the energies ε  10, the calculations are performed by means of Eqs. (18), (24) and (25). The dimensionless energy ε is defined as ε = E / I , where E is the energy of the incident electron and I = η2 /(2m) is the Coulomb ionization potential for a K-shell electron

ε

Q (ε )

Z 4 σK+

1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.25 3.5 3.75

0.4673 0.7284 0.8879 0.9842 1.0391 1.0674 1.0784 1.0779 1.0695 1.0553 1.0381

41.108 64.079 78.112 86.584 91.410 93.902 94.871 94.825 94.088 92.843 91.324

ε 4 .0 4.25 4 .5 4.75 5 .0 5 .5 6 .0 7 .0 8 .0 9 .0 10

Q (ε )

Z 4 σK+

ε

Q (ε )

Z 4 σK+

ε

1.0184 0.9978 0.9764 0.9546 0.9327 0.8901 0.8624 0.8085 0.7557 0.7041 0.6540

89.592 87.781 85.897 83.981 82.056 78.302 75.865 71.124 66.485 61.940 57.538

15 20 25 30 35 40 45 50 55 60 65

0.4913 0.3940 0.3298 0.2841 0.2500 0.2235 0.2023 0.1850 0.1705 0.1582 0.1476

43.223 34.665 29.011 24.996 21.995 19.664 17.799 16.271 14.996 13.915 12.986

70 75 80 85 90 95 100 105 110 115 120

Q (ε )

Z 4 σK+

ε

Q (ε )

0.1384 0.1304 0.1233 0.1169 0.1112 0.1061 0.1014 0.0971 0.0933 0.0897 0.0864

12.179 11.471 10.844 10.285 9.784 9.332 8.921 8.547 8.204 7.889 7.599

125 130 135 140 145 150 155 160 165 170 175

0.0833 0.0805 0.0778 0.0754 0.0731 0.0709 0.0689 0.0670 0.0651 0.0634 0.0618

Z 4 σK+ 7.330 7.080 6.848 6.631 6.428 6.238 6.059 5.891 5.732 5.582 5.440

ε

Q (ε )

180 185 190 195 200 210 220 230 240 250 300

0.0603 0.0588 0.0575 0.0561 0.0549 0.0526 0.0504 0.0485 0.0467 0.0450 0.0382

Z 4 σK+ 5.305 5.177 5.056 4.940 4.830 4.624 4.436 4.263 4.104 3.956 3.360

mensionless units, the energy-conservation law now implies ε − μ = ε1 + με2 , where μ = m/ M and ε2 = E 2 / I is the dimensionless energy of the ionized electron. The Born approximation is valid within the non-relativistic energy range μ  ε  2(α Z )−2 . It can be easily seen that the formulas for the differential cross section dσK+ given by Eqs. (17)–(21) can be also applied to the single ionization of a K-shell electron by impact of arbitrary charged particles. However, the integration limits x1,2 now should be deformed by the mass ratio μ as follows x1 = x2 =

Fig. 3. The universal quantity Q (ε ) calculated as a function of the dimensionless energy ε . Dotted line, calculation within the Born approximation according to Eqs. (18) and (25); solid line, exact calculation according to Eqs. (53) and (55).

q2min

η2 q2max

η

2

=

2

μ2 x2

= μ−2

(27)

,

√

√

2

ε + ε − μ ,

(28)

where  = 1 + ε2 . If μ = 1, the total cross section σK+ is given by Eqs. (24) and (25), but the integration over the variable ε2 should now be performed within the scaled energy range 0  ε2  (ε − μ)/μ. We have performed numerical calculations of the scaling functions Q (ε ) for particular examples of the K-shell electron ionization by (anti)muon and (anti)proton impacts. The integral over ε2 is rather unsensitive to the extension of the integration range due to deformations by the mass ratio μ, being saturated near the lower limit. Since μ  1 in the cases considered, the difference between the scalings for these heavy projectiles is not visible (see Fig. 4). In addition, at asymptotic high-energy limit, the scaling functions Q (ε ) for different incident particles are degenerated into the universal function of the dimensionless energy ε , which already depends on neither the charge nor the mass of the incident particle [3,4]. 3. Now we shall consider the near-threshold energy domain in more details. In this case, the Born approximation should not be adequate. Both diagrams depicted in Figs. 1(a) and (b) give comparable contributions to the ionization cross section and should be taken into account. Accordingly, the total amplitude of the process reads

Fig. 4. The scaling functions Q (ε ) for the K-shell electron ionization by different charged projectiles. The calculations are performed within the Born approximation. Dotted line, the electron impact; dashed line, the positron impact; solid line, impact by a heavy particle with the mass M  m.

law has the same form E − I = E 1 + E 2 , where E = M υ /2. However, now it is convenient to calibrate the energies of the incident and scattered particle by the characteristic binding energy ˜I = M (α Z )2 /2, namely, ε = E / ˜I and ε1 = E 1 / ˜I . According to this definition, the dimensionless energy ε = υ 2 /(α Z )2 does not depend on the mass of the incident particle. In di2

A = Aa δτ1 τ1 δτ2 τ2 − Ab δτ2 τ1 δτ1 τ2 ,

(29)

Aa = ψ p 1 ψ p2 | V 12 |ψ p ψ1s ,

(30)

Ab = ψ p2 ψ p1 | V 12 |ψ p ψ1s .

(31)

Here τ1,2 and τ1 ,2 denote the spin projections of the Pauli spinors

in the initial and final states, respectively. The Coulomb interaction between two electrons is described by the operator V 12 = α /|r 1 − r 2 |, where r 1 and r 2 are the electron coordinates. In the momentum representation, the amplitude Aa is given by Eq. (1). The amplitude Ab originates from the exchange diagram.

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Fig. 5. The universal function Q (ε ) is compared with the experimental data for ionization of hydrogen-like ions by electron impact. Dotted line, calculation within the Born approximation according to Eqs. (18) and (25); solid line, exact calculation according to Eqs. (53) and (55). Experimental data: H atom,  [22]; He+ ,  [23]; Li2+ ,  [24]; B4+ ,  [25]; C5+ , [25], # [26]; N6+ ,  [25],  [26].

In this paragraph, the calculations are carried out in the coordinate space. As the wave functions of the initial particles, we shall take the corresponding solutions of the Schrödinger equation for an electron in the external field of the Coulomb source [17]

ψ1s (r ) = N 1s e −ηr , ψ p (r ) =

∞ 4π

2p

l=0

l i δ pl

ie

The asymptotical behavior of the wave function ψ p (r ) is “the sum of a plane wave and a spherically outgoing one”. The functions (33) are normalized by the condition



dr ψ p∗ (r )ψ p (r ) = (2π )3 δ( p  − p ).

(35)

(32) R pl (r )

l

For the attractive Coulomb field of a point nucleus, one has [17] ∗

ˆ ). Y lm (ˆr )Y lm ( p

(33)

2 = η3 /π , η = mα Z ,1 Y lm (ˆr ) are the spherical harmonics Here N 1s depending on the variable rˆ = r /r, and δ pl are the phase shifts of the radial functions R pl . The latter are orthogonal and normalized according to

∞

dr r 2 R p  l (r ) R pl (r ) = 2π δ( p  − p ).

(34)

0

1 The use of the same notations for the electron mass and projection of the orbital angular momentum is not confusing, because the latter is related with the angular dependence of wave functions only.

C pl

(2pr )l e −ipr Φ(l + 1 + i ξ, 2l + 2, 2ipr ), (2l + 1)!



C pl = 2pe π ξ/2 (l + 1 − i ξ ) ,

(36)

δ pl = arg (l + 1 − i ξ ),

(38)

R pl (r ) =

m=−l

(37)

where ξ = η/ p, Φ(a, b, z) is the confluent hypergeometric function, and ( z) is the Euler’s gamma function. Outgoing electrons may be described by the wave functions of the stationary states characterized by the definite values of the energy, the angular momentum, and its projection, namely,

ψ p 1 (r ) = R E 1 l1 (r )Y l1 m1 (ˆr ),

(39)

ψ p 2 (r ) = R E 2 l2 (r )Y l2 m2 (ˆr ).

(40)

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Fig. 6. As Fig. 5, for ionization of hydrogen-like ions by electron impact. Experimental data: O7+ ,  [25], [26]; Ne9+ , # [26]; Ar17+ , [26]; Fe25+ ,  [27]; Mo41+ ,  [28], ♦ [29]; Dy65+ , [28].

mα l i δ pl l i e V l2 l1 C ll01 m1 l2 m2 , √ η2 p

The radial functions R El (r ) are normalized to δ function in the energy

Ab =

∞

W ll1 l2 = √

dr r

2

R E l



(r ) R El (r ) = δ( E − E ),

(41)

being related to R pl (r ) as follows

√ m R pl (r ). R El (r ) = √

The partial-wave decomposition of the Coulomb interaction V 12 is given by [19] V 12 =

λ=0

4πα

λ r<

λ+1 (2λ + 1) r>

λ

Y λ∗μ (ˆr 1 )Y λμ (ˆr 2 ),

(43)

μ=−λ

where r< = min{r1 , r2 } and r> = max{r1 , r2 }. Choosing the z-axis along the momentum p, we can perform the integrations over the angular variables rˆ 1 and rˆ 2 in the matrix elements (30) and (31). It yields

Aa =

I ll1 l2 =

mα

√

η2 p

l

(46)

1

Πl2

π kk1 k2 Πl1

C ll01 0l2 0 J ll2 l1 ,

(47)

(44)

∞

0

J ll2 l1

x<2

√

l2 +1

l

dx2 x22 R k1 l1 (x2 ) 0

e −x2 ,

(48)

e −x2 ,

(49)

x>

∞

dx1 x21 R k2 l2 (x1 ) R kl (x1 )

=

l

dx2 x22 R k2 l2 (x2 ) 0

∞

x<1 l +1

x>1

where Πl = 2l + 1, x< = min{x1 , x2 }, x> = max{x1 , x2 }, and C llmm l m denotes the Clebsch–Gordan coefficient. In Eqs. (46)–(49), 1 1 2 2 we have introduced dimensionless momenta k = p /η , ki = p i /η and dimensionless coordinates xi = ηr i (i = 1, 2). Accordingly, the radial functions (36) satisfy to the relation R pl (r ) = η R kl (x). Due to identity of the electrons, the functions V ll l can be obtained from W ll

1 l2

il e i δ pl W ll1 l2 C ll01 m1 l2 m2 ,

C ll01 0l2 0 I ll1 l2 ,

dx1 x21 R k1 l1 (x1 ) R kl (x1 )

0

∞

Πl1

∞ (42)

2π p

1

π kk1 k2 Πl2

V ll2 l1 = √

0

(45)

l

2 1

by simultaneous replacements k1  k2 and l1  l2 .

The differential cross section dσK+ is given by the expression (9), where the phase volume dΦ is equal to dΦ = dE 1 dE 2 .

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Fig. 7. The universal function Q (ε ) is compared with the experimental data for the single K-shell ionization of helium-like ions by electron impact. Dotted line, calculation within the Born approximation according to Eqs. (18) and (25); solid line, exact calculation according to Eqs. (53) and (55). The experimental measurements are interpreted according to Eq. (56). Experimental data: B3+ ,  [32]; C4+ ,  [26],  [32]; N5+ , [26], # [32]; O6+ , [26]; Ne8+ ,  [26];  [33]; Ar16+ , [26].

In addition, it should be averaged over the polarizations of the initial electrons and summarized over the polarizations of the final electrons. This can be achieved by means of the following substitution 1 | A| 2 → | A| 2 = | A| 2 . (50) 4   l1 ,m1 l2 ,m2

τ1 , τ1 τ2 , τ2

Note that the summations are also performed over the angular momenta of the outgoing electrons, because their states are not fixed. Then the differential cross section takes the form dσK+ dE 1

= 2π

m p



|Aa |2 + |Ab |2 −

l1 ,m1 l2 ,m2

1 2

Aa Ab∗ + Aa∗ Ab





.

(51)

Let us again introduce the dimensionless energies ε = E / I and εi = E i / I (i = 1, 2), where I = η2 /(2m) is the ionization potential for the K-shell electron. In addition, we shall also assume that ε1  ε2 , that is, (ε − 1)/2  ε1  ε − 1. Inserting Eqs. (44) and (45) into Eq. (51) yields dσK+ dε1

=

σK+ =

σ0 Z

F (ε , ε1 ), 4

F (ε , ε1 ) =

1 

ε

l,l1 ,l2

where σ0 = π a20 and a0 = 1/(mα ). Similarly to Eq. (52), one could write the energy distribution for slowly escaping electrons, which is described by the universal function F (ε , ε2 ) within the energy range 0  ε2  (ε − 1)/2. In Fig. 2, the function (53) is calculated for few values of the dimensionless energy ε . In the vicinity of the threshold, the energy dependence turns out to be rather weak. The comparable contributions to the total cross section arise from the emission of electrons with arbitrary energy sharing. In addition, the predictions for the energy distributions according to Eq. (53) deviate strongly from those calculated within the Born approximation without taking into account the electron–electron exchange interaction. The latter neglects the electron–nucleus interaction, which is especially crucial in the regime of almost equal-energy sharing ε1  ε2 , because in this case the nucleus takes significant part in the recoil momentum transfer. The total cross section for the single K-shell ionization reads

2 W ll1 l2

(52)

+



2 V ll2 l1

−

W ll1 l2 V ll2 l1



,

(53)

σ0 Z4

Q (ε ),

(54)

ε−1 

Q (ε ) = (ε −1)/2

F (ε , ε1 ) dε1 ,

(55)

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Fig. 8. The universal function Q (ε ) is compared with the experimental data for the single K-shell ionization of neutral atoms by electron impact. Dotted line, calculation within the Born approximation according to Eqs. (18) and (25); solid line, exact calculation according to Eqs. (53) and (55). The experimental measurements are interpreted according to Eq. (56). Experimental data: C,  [34,35]; Si,  [34,36]; K, [34,37]; Ca,  [34,37]; Cr,  [38]; Mn, # [39].

where F (ε , ε1 ) is given by Eq. (53). It should be stressed that the universal scaling Q (ε ) is obtained, taking into account the leading order of non-relativistic perturbation theory. The correlation corrections of the order of 1/ Z and the relativistic corrections of the order of (α Z )2 [20] have been omitted in the present consideration. Eqs. (54) and (55) are valid for hydrogen-like ions with moderate values of Z . We have calculated the function Q (ε ) according to Eqs. (53) and (55) for the electron energies within 1 < ε < 10 (see Table 1 and Fig. 3). Near the ionization threshold, the exact function Q (ε ) exhibits a more pronounced maximum rather than the prediction (25) obtained by using the Born approximation, although the deviation is not too significant. Account of the exchange interaction and the attractive field of the atomic nucleus result in the shift of the maximum of the curve. The exact function Q (ε ) reaches its maximum at lower energy ε  2.85, while the Born approximation predicts ε  3.45. Surprisingly enough, that the total cross section exhibits the asymptotic high-energy behavior already at relatively low energies ε  5 (see also Ref. [21]). For energies within the range 1  ε  2, the function (55) can be approximated by a Taylor expansion of the following form:

Q (ε ) = n1 B n (ε − 1)n . The first four coefficients turn out to be B 1 = 2.53, B 2 = −3.20, B 3 = 2.55, and B 4 = −0.90. The coefficient B 1 gives the slope of the curve Q (ε ) at the threshold point ε = 1.

In Figs. 5 and 6, the universal function Q (ε ) is compared with available experimental data for hydrogen-like targets. In the case of light atomic systems, the deviation of experimental results from the curve is due to higher-order correlation contributions, which are most prominent in the case of the hydrogen atom and the He+ ion. However, at high non-relativistic energies, the two-photon exchange diagrams appear to be unimportant. For example, for the H atom, the Bethe asymptotic scaling (26) is reproduced experimentally at ε  50 [22], while the measurements for the He+ ion are in agreement with the theoretical curve already at about ε  15 [23]. With increasing values of Z , the experimental data rapidly approach the universal curve Q (ε ) within the near-threshold energy domain. For heavy multicharged ions with α Z  1, the relativistic effects become increasingly important [11]. The latter may explain the difference between our predictions and experimental results for the Dy65+ ion. Note also that, in this case, the energies of the incoming electron have been calibrated by the experimental value for the ionization potential. If the K shell is filled, the electron wave functions possess a non-Coulomb behavior, while the Coulomb potential I does not correspond to the ionization threshold. The screening effect can be simulated by replacing the true nuclear charge Z by an effective value Z eff . The latter can be defined by equating the experimental potential for the single ionization of a K-shell electron and the

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Table 2 For various atomic targets, the nuclear charges Z , the Coulomb ionization potentials I = m(α Z )2 /2, the effective nuclear charges Z eff , and the experimental potentials I exp for the single K-shell ionization [31] are tabulated Target

Z

I (keV)

Z eff

I exp (keV)

Target

Z

I (keV)

Z eff

I exp (keV)

Target

Z

I (keV)

Z eff

I exp (keV)

B3+ O6+ C Ca Fe Ga Rb Ba

5 8 6 20 26 31 37 56

0.340 0.871 0.490 5.442 9.197 13.075 18.626 42.667

4.366 7.371 4.664 17.249 22.871 27.609 33.427 52.458

0.259 0.739 0.296 4.048 7.117 10.371 15.203 37.441

C4+ Ne8+ Si Cr Ni Ge Sr La

6 10 14 24 28 32 38 57

0.490 1.361 2.667 7.837 10.667 13.932 19.647 44.205

5.368 9.376 11.642 20.991 24.755 28.572 34.408 53.488

0.392 1.196 1.844 5.995 8.338 11.107 16.108 38.925

N5+ Ar16+ K Mn Cu As Sb Pr

7 18 19 25 29 33 51 59

0.667 4.408 4.912 8.504 11.442 14.817 35.388 47.361

6.370 17.403 16.298 21.931 25.699 29.538 47.344 55.554

0.552 4.121 3.614 6.544 8.986 11.871 30.496 41.991

Fig. 9. As Fig. 8, for the single K-shell ionization of neutral atoms by electron impact. Experimental data: Fe,  [40].

effective one, that is, I exp = m(α Z eff )2 /2 [30] (see Table 2). Accordingly, the energy of incoming electron E is measured in units of the experimental value I exp . The total cross section for the single K-shell ionization now reads as follows

σK+ =

2σ0 4 Z eff

Q (ε ),

(56)

where σ0 = π a20 , a0 = 1/(mα ), and the function Q (ε ) is again given by Eqs. (53) and (55). The factor 2 accounts for the number of electrons in the closed K shell. Fig. 7 compares the universal scaling Q (ε ) with the experimental data for helium-like ions, while Figs. 8–10 do it for neutral atoms. The agreement between

[39]; Ni,  [38]; Cu,  [38]; Ga,  [40]; Ge, ♦ [41]; As,

the results is quite satisfactory, although the measurements reported in the original work [37] seem to be unreliable near the ionization threshold. Concluding, we have deduced the universal scaling behavior for cross sections of the single K-shell ionization by electron impact. The results are obtained within the framework of nonrelativistic perturbation theory, taking into account the one-photon exchange diagrams. The scaling law turns out to be applicable both for multicharged ions and neutral atoms. The accurate check of the universal scaling behavior of the ionization cross sections requires new experimental measurements with higher precision. The hydrogen-like multicharged ions with moderate values of the

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Fig. 10. As Fig. 8, for the single K-shell ionization of neutral atoms by electron impact. Experimental data: Rb, # [34,37]; Sr, [34,43]; La,  [34,43]; Pr,  [34,43].

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