Theoretical Models of Inner-Shell

Transcription

1 Chapter 2 Theoretical Models of Inner-Shell Ionisation Several theoretical methods have been developed to describe the process of innershell ionisation in heavy ion-atom collisions by utilizing different approximation methods. The theories describe a systematic expositions of the various mechanisms governing the inner shell vacancy production in ion atom collisions. The information about various microscopic and macroscopic aspects of ion-atom combinations is provided through ion-atom collision. In ion-atom collision, inner shell vacancy can be created by various processes and has a fundamental interest from the point of view of quantum mechanical scattering theories. The ion induced inner shell excitations have provided an enormous amount of input for the evolution of quantum theory in its early growth period [1]. Consequently, this chapter offers a general view for the main concepts of calculating the ionisation cross section theoretically. 11

2 2.1. Binary encounter approximation 2.1 Binary encounter approximation The binary encounter approximation (BEA), which is also known as impulse approximation, assumes that ionisation is completely due to a classical binary encounter between the charged projectile and the target electron. The target nucleus and the rest of its electrons are assumed to play no role in the process, except for providing the initial momentum distribution and binding energy of the ejected electron. A detailed analysis of the classical Coulomb interaction between two moving charged particles was given by Gryzinski [2 4]. Rudd et al. [5] extended the theory by using a quantum-mechanically derived velocity distribution for the target electron. The expression for the K-shell ionisation cross section [6] is given by, σ k (E i ) = N kz1σ 2 0 G(V ), (2.1) U2k 2 where U 2k is the electron binding energy, N K is the number of electrons in the K-shell and σ 0 is given by, σ 0 = πe 4 = cm 2 ev 2 (2.2) The term G(V ) is a function of the reduced velocity v 1 /v 2k = V, and it is given in references [2 4, 6]. The BEA predicts that the product of the binding energy squared and the ionisation cross section divided by the projectile atomic number squared is a universal function of the reduced velocity V. The utility of the BEA calculations are their simple character and the universal scaling laws that are the result of using appropriate hydrogenic wave functions to obtain the K- shell electron velocity distributions. However, it is to be noted that U 2k is related to the 1s velocity distributions and hence the BEA calculations are appropriate only to K-shell ionisation processes. In the case of L-subshell ionisation it is not 12

3 2.2. Semi-classical approximation appropriate to simply scale the BEA results for K-shell ionisation. 2.2 Semi-classical approximation The semi-classical approximation (SCA) [7 10] was introduced by Bang and Hansteen [7], with the aim to study the effects of projectile deflection and deceleration due to atomic Coulomb excitation during collision by light ions, Z 1 Z 2. Here, Z 1 and Z 2 represent the atomic numbers of the projectile and target atoms respectively. The necessary condition for the ionisation process to be treated classically is that, the de-broglie wavelength of the moving ion is smaller than the distance of closest approach i.e. 2d λ = 2 Z 1 Z 2 e 2 h v (2.3) This ratio is called Coulomb parameter. Here d is the half distance of the closest approach in a head-on collision and is given by, d = Z 1 Z 2 e 2. (2.4) M 1 v1 2 Here λ is the de-broglie wavelength of the projectile. The SCA enables one to investigate the details of the collision process as a function of impact parameter. It can be derived from the basic principles of quantum mechanics (first-order timedependent perturbation theory) in a relatively straight-forward way. so that the number of parameters introduced in a more or less artificial manner is minimised. The Coulomb repulsion between the projectile and the target is taken in to account by assuming a hyperbolic trajectory for the incident projectile. The differential cross section for the ionisation of an atomic electron with a final energy E f, as given by first-order time-dependent perturbation theory, is 13

4 2.2. Semi-classical approximation [7, 11], dσ = 2π de f 2 0 b db e iωx ψ f V (r, t) ψ i dt 2. (2.5) Where b is the impact parameter and ω = (E f + u i )/( ) with u i being the binding energy of the electron in the initial bound state, and E f denoting the final state energy of the electron. The quantity V (r, t) is the time-dependent Coulomb potential between the projectile and the target electron, and ψ i and ψ f are the one-electron states in the self-consistent field, as only one-electron excitation is possible in the first order theory. The ionisation probability, I b, as a function of impact parameter b is thus given by, where I b = E max 0 [ di(b) de f ] de f, (2.6) di(b) de f = a Efi (t ). (2.7) where, a Efi (t ) is an excitation amplitude given by [8], a Efi (t ) = i dt e iωx Z 1 e 2 E f r R b (t) i. (2.8) Tabulation of the matrix elements in Equation (2.7) is given in Ref. [7]. Instead of using hydrogen-like wave functions for the bound electron, Trautmann et. al. [9, 12 14] employed relativistic Hartree-Fock electron wave function. They also made corrections for binding and polarisation effects, nuclear distortions, screening on projectile trajectory, and recoil effects. Reviews and corrections of the SCA are available in references [7, 9, 12 17] and the earlier references given therein. Those corrections have been shown to significantly improve agreement between the theory and experiment. 14

5 2.3 Plane wave Born approximation 2.3. Plane wave Born approximation Plane wave Born approximation (PWBA) is a quantum perturbation method in which the first Born approximation (FBA) is used to describe the interaction between the projectile and the target. This method is generally valid when [1, 18], Z 1 e 2 hv 1 = 1. (2.9) Where Z 1 is the projectile atomic number, v 1 is the projectile velocity, and e 2 /h is the speed of the electron in a hydrogen atom. In addition to the condition of Equation (2.9), it is assumed that i) the interaction between the projectile and the target electron is very weak, ii) the target electron appears frozen during the collisions because the response time of the electrons is long compared to the interaction time, and iii) the projectile acts as a point charge and its electronic structure has a negligible effect on the interaction [19]. Generally, these conditions are fulfilled for Z 1 Z 2 and v 1 v 2k. The scattering amplitude in this approximations for transition from state n to m is given by; f = m e i K R V e i K R V n, (2.10) where V is taken to be the Coulomb potential between the projectile and the electron initially in the state n. The quantities K and K denote the initial and final relative momentum. Details of the evaluation of the cross section for ionisation employing the PWBA are available in reference [20]. The result for K-shell ionisation is given by the formula [1, 18 20], σ P W BA K = (σ 0K /θ K )F K ( η K, θ θk 2 K ), (2.11) 15

6 2.3. Plane wave Born approximation where σ 0K is σ 0K = 8πa 2 0(Z 1 /Z 2 2K), (2.12) in which a 0 = Å is the Bohr radius, and Z 2K = Z is the screened target nuclear charge. In Equation (2.11), θ K measures how much the K-shell ionisation energy exceeds that of a hydrogenic atom and is given by θ K = U K /Z 2 2KR E. (2.13) Where U K is the K-shell binding energy and R E = 13.6 ev is the Rydberg energy. The reduced particle-velocity parameter, η K, in Equation (2.11) is given by, η K = ( v1 v 2K ) 2 (2.14) Extensive tables of values of the function F K in Equation (2.11) have been tabulated for the L-shells by Khandelwal and Choi [21, 22]. PWBA is in good agreement with experiment for light projectiles at high velocities. For high projectile energies, PWBA is equivalent to SCA, as the straight line trajectory for the projectile ion used in the former becomes a reasonable approximation at higher energies [7, 11, 17, 23]. Complications arise when heavy ion projectiles are used and in case of slow collision conditions. In these conditions it is found that the data for both the K and L shell ionisation are no longer in agreement with the predictions of the theory. This is due to modification of the binding energy of the bound electrons of the target in presence of the projectile ion, Coulomb deflection of the projectile due to the target nucleus and enhanced cross section for electron transfer from the bound state of target atom to the bound state of projectile ion. However, various improvements have been suggested to overcome the limitations of this theory. Examples are (i) Distorted Wave Born Approximation (DWBA) (see a review by 16

7 2.4. Perturbed stationary state approximation Rudd et al. [24]) that includes the interaction between the projectile and the target nucleus, (2) Continuum-Distorted-Wave method (CDW) [12, 13] in which Coulomb interactions are explicitly contained in the initial- and final-state wave functions, etc.. In the PWBA theory, corrections for Coulomb repulsion [25], projectile energy loss [23, 26, 27], relativistic effects [10, 20, 23, 28, 29], and binding polarisation effects [19, 25, 27] led to the well-known ECPSSR theory. 2.4 Perturbed stationary state approximation The ECPSSR theory goes beyond the first-order perturbative treatment and is the most advanced approach based on the PWBA. It includes corrections for projectile energy loss (E), Coulomb deflection of the projectile trajectory (C), binding and polarisation effects within the perturbed stationary state (PSS) of the target electron, and relativistic (R) effects. The Coulomb deflection correction is very important for low energy projectiles. The binding effect refers to the increased binding experienced by the electron at small impact parameters when it is in the combined field of the projectile and target nuclei. The polarisation effect refers to the distortion of the electron wave function by the projectile at impact parameters greater than the electron shell radius. In addition to the above mentioned corrections, ECPSSR also takes into account vacancy production via electron capture to the projectile based on the Oppenheimer- Brinkman-Kramers (OBK) formula [30 33]. In slow collision limits, the calculations of first Born approximation (FBA) scales with the central dimensionless parameter ξ as, ξ s = 2v 1 = 2nη1/2 s (2.15) v 2s θ 2s θ s In ECPSSR theory [34], ξ s is replaced by ξ R s /ζ s to correct the first Born approx- 17

8 2.4. Perturbed stationary state approximation imation for the relativistic and perturbed stationary state effects. Here ξ R s = [ m R s (ξ s /ζ s ) ] 1/2 ξs to simulate the relativistic effect and ζ s accounts for the PSS effect. Also, θ s is replaced by ζ s θ s. The relativistic correction in the electron mass m R s can be expressed in atomic units as; m R s (ξ s ) = ( y 2 ) (1/2) + y, (2.16) with y = 0.4 ξ s ( ) 2 Z2s. (2.17) 137 Hence, the ECPSSR ionisation cross section is formulated in terms of σ P W BA s Equation (2.11) as [34, 35], ( ) ECP SSR 2π d q0s ξ s σs = C s z s (1 + z s ) from ( ) ξ f s (z s ) σs P W BA R s, ζ s θ s. (2.18) ζ s Where C s represents the correction for Coulomb deflection, the quantity d = Z 1 Z 2 /(M 1 v1) 2 is the half distance of the closest approach in a head-on collision, q 0K represents the approximate minimum momentum transfer (U 2s /v 1 ), f s (z s ) stands for the energy loss correction and z s is given by, z 2 s = 1 4 Mζ s θ s ( ζs ξ s ) 2. (2.19) The projectile energy loss during the collision process (f s (z s )) is given by, f s (z) = 1 ( ) 45 ((1 z 2 ) (1 z 2 ) 2). (2.20) 16 The term ζ used to represent perturbed stationary state effects can be expressed as, ζ s = 1 + Z 1 Z 2s θ s [g(ξ s ) h(ξ s )] (2.21) 18

9 2.4. Perturbed stationary state approximation Detailed calculation procedure for the ECPSSR cross section can be found in references [35, 36]. It has been shown that ECPSSR calculations agree with experimental results within 10% to 20% for proton and alpha particle bombardment of targets in the range 10 Z2 92 [37]. But in case of heavy ion projectiles when Z 1 Z 2, and v 1 /v 2s 1 is no longer true, there is considerable difference between the experimental data and ECPSSR predictions. Furthermore, when the energy of the projectile decreases below 1 MeV/u, the ECPSSR theory, along with other theories, fail to provide good predictions for the L shell ionisation cross section even for the collisions by lightest projectile ions i.e. protons [38] United Atom formalism The ECPSSR cross sections are calculated by an appropriate scaling of the PWBA cross sections evaluated with the non relativistic screened hydrogenic (SH) wave functions and with a minimum momentum transfer q 0s that neglects the energy loss effect. Chen and Crasemann produced PWBA cross sections calculated with relativistic Dirac Hartree Slater (DHS) wave functions and with the exact q 0s [39, 40]. They observed that a wave function factor, W σ EP W BAR s EP W BAR (DHS)/σs (SH), (2.22) by which the ECPSSR based on SH wave functions should be multiplied to account for a switch to a DHS description of the target atom. In the ECPSSR, θ s is modified to ζ s θ s where the PSS function (Equation (2.21)) was derived in the Separated Atom (SA) picture as a first-order perturbation of the S-shell electron binding energy; the 0 < g s (ξ s ) 1 accounts for the increase in the binding that increases with decreasing ξ s as the projectile penetrates deeper into the S shell and h s (ξ s ) accounts for the decrease in the binding at intermediate ξ s where the 19

10 2.4. Perturbed stationary state approximation projectile polarises the S shell in its passage outside the S shell. In the slow collision limit of ξ s 0, with h s = 0 and g s 1, the PSS function becomes ζ s = 1 + 2Z 1 /Z 2s θ s. Therefore, the binding energy is replaced with the binding energy of the united atom (UA) i.e. 1(Z Z 2s ) 2 θs UA /n 2, so that the ζ s becomes, ( ζs UA = 1 + Z ) 2 1 θs UA. (2.23) Z 2s θ s ζs UA truncates the increase of the binding energy in theξ s 0 limit. Since the decrease in the binding energy yields greater cross sections, the ionisation cross sections in low energy region evaluted with ζ UA s is higher than the ECPSSR cross sections. Here, the ECPSSR model is modified to ECUSAR model via the ζ s ζ USA s replacement where, ζ USA s = ζ UA s when ζ UA s ζ s for slow collisions (2.24) ζ s when ζ s ζ UA s for intermediate and fast collisions This joins the united and separated atom formulae as they are derived and valid in the complementary collision regimes. This modification brings the theory to a closer agreement with the data; although it lowers difference by just a few percent for heavy targets, it cuts the difference between the data and the theory in half for the light targets [41]. Still more stringent test of the theory remains as there is lack of precise data for the L shell ionisation. 20

11 2.5. Geometrical model 2.5 Geometrical model For heavy projectiles, the BEA and the SCA theories fail since their predictions of ionisation probabilities exceed unity. The geometrical model (GM) [42 44] was developed to describe multiple ionisation in heavy ion-atom collisions, involving strong perturbations. Within the framework of the single particle model (SPM) [45], the simultaneous inner and outer shell ionisation processes are characterised by the inner shell ionisation cross section and the ionisation probability per electron for the outer shells at zero impact parameter. Details of the geometrical model (GM) are given in references [42 44]. The projectile-target nucleus impact parameter is taken to be zero for inner-shell ionisation while the distance between the projectile trajectory and the electron is denoted by b. The fraction of the electrons swept out by the projectile may be considered as the ionisation probability per electron. With the assumption that, the dependence of the efficiency of this ejection on the distance b measured from the projectile trajectory, the ionisation probability P nlm (b) on the basis of a descriptive geometrical picture is given by, P nlm (b) = ψ nlm ( r) 2 η(b)ρ dρ dz dφ. (2.25) where r is the position vector of the electron, ψ nlm is the wave function of the electron with quantum numbers n, l and m. ρ dρ dz dφ is the differential volume in cylindrical coordinates, and η(b)is the efficiency function, which is given by, η(b) = 0 if b > b 0 1 if b b 0 (2.26) 21

12 2.5. Geometrical model with b 0 given by the ionisation cross section b 0 = [ σrt π ] 1/2 = Z 1 v 2 1 [ ] 2v 2 1/2 1 I 1 Z 1 v 1 [ ] 1/2 2, (2.27) I in atomic units, with Z 1 and v 1 being the projectile charge and velocity respectively. The experimental binding energy is given by I and σ RT represents the Rutherford-Thomson cross section. This approximation is considered to work in high velocity region of v 1 v 2s. If σ RT is replaced with σ BEA i.e. the BEA cross section of ionisation, then, b 0 = Z 1 v 1 V [ ] 2 2G(V ) (2.28) I where V = v 1 /v 2 is the reduced projectile velocity, G(V ) is the universal BEA function [6], and I is the experimental binding energy given by v 2 2/2. In order to develop a practical formula, an universal variable X is defined as, X = a nl b 0, (2.29) where, a nl = 2 Z 2 n = 2v 2, (2.30) is an atomic units. From above three equations, the universal variable X, which may be interpreted as the measure of the perturbation strength characterising the collision, is written as, X = 4 Z 1 v 1 V [G(v)] 1/2. (2.31) For the different n, l subshells the corresponding G nl (V ) functions are used as G(V ). The mean ionisation probability per electron at zero impact parameter for higher shells may be described approximately by introducing the universal 22

13 2.5. Geometrical model scaling parameter, X n = X/n. It has been shown that, the mean ionisation probability per electron at zero impact parameter depends only on the universal scaling parameter X n which involves the mean quantum number n of the given shell [44]. This simple rule makes it possible to estimate quickly the ionisation probabilities by the approximation formula given by; P n (0, X n ) = X 2 n X 2 n[ exp( X 2 n/16) (2.32) In summary, the Geometrical Model predicts that the mean ionisation probability per electron at a zero impact parameter depends only on the universal scaling parameter X. 23

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