ELASTIC SCATTERING OF X-RAYS AND Γ-RAYS BY 2s ELECTRONS IN IONS AND NEUTRAL ATOMS *

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1 Romanian Reports in Physics, Vol. 64, No. 4, P , 0 ELASTIC SCATTERING OF X-RAYS AND Γ-RAYS BY s ELECTRONS IN IONS AND NEUTRAL ATOMS * K. KARIM, M. L. MUNTEANU, S. SPÂNULESCU,, C. STOICA University of Bucharest, Department of Physics, MG, Bucharest-Magurele 0775, Romania Hyperion University of Bucharest, Department of Physics, Bucharest 03069, Romania Received October 7, 0 Abstract. A dynamic screening model based on nonrelativistic Hartree-Fock approach is given for the s subshell Rayleygh scattering amplitudes and the total photoect cross-section. It considerably improves the predictions for low atomic number elements especially in a large energy range starting from one and half the photoect threshold energy. A good agreement with full relativistic numerical calculations is observed for these elements and energies also. Key words: dynamic screening model, elastic scattering, X-rays and Γ-rays.. INTRODUCTION Since long time ago the subject of X-rays and Γ-rays scattering by inner shell electrons in atoms has presented a large interest. Experimental and theoretical advances have allowed a continuous progress in this topic, the results of such investigations being useful in many fields such as laser thermonuclear fusion, plasma physics, lasers, astrophysics and others. As important experimental progress, we want to mention the creation of the X-ray free electron laser with high spectral resolution [,, 3], while as theoretical results we mention the works [4 0]. Usually the term inner shell electrons is used to describe in a fairly good approximation electrons whose quantum states given by wave functions, energies, orbital kinetic moment etc. are perturbed very little by the interactions with the other electrons in the atom. More specific, it is realistic to consider that K-shell and L-shell electrons may be described by hydrogen-like wave functions and energies with an appropriate ective charge Z. In such a way, photon scattering by s and s bound electrons may be described by analytical formulae which give precise numerical predictions. Further, it is realistic to consider that inner electrons hydrogen-like quantum states are less perturbed by interatomic interactions in molecules and solids. * Paper presented at the Annual Scientific Session of Faculty of Physics, University of Bucharest, June 7, 0, Bucharest-Magurele, Romania.

2 Elastic scattering of X-rays and Γ-rays 987 It has been proven [8 0] that the Coulombian shape of inner s and s electrons wave functions with an ective charge Z obtained by fitting the Hartree-Fock charge distribution with the Coulombian one is a good approach which gives very good predictions for Rayleigh and photoect cross-sections. As expected, this simple screening model is especially good in the case of large atomic numbers Z (when the interaction of the inner electron with the nucleus is strong) and far enough from the photoect threshold (at least one and a half the photoect threshold energy). In this paper we are able to improve our screening model considering a dynamic screening constant, i.e. in the case of large photon energies the ective charge is the same for a large energy range as it is given by the Hartree-Fock calculations, but below twice the ionization energy we consider th that Z decreases linearly till a threshold value Z given by the experimental threshold energy th th + α Z ω s = ( Z ) m, where ( Z ) =. Thus we follow the same approach that we used in a previous paper [] in the case of Rayleigh scattering by K-shell electrons. We want to mention that in the case of s electrons, the screening model must be considered in a more subtle way, because the ective charge variation is significantly larger that in the K-shell case and the charge distribution spans a larger range. /. THE DYNAMIC SCREENING MODEL FOR RAYLEIGH SCATTERING AND PHOTOEFFECT FOR THE s SUBSHELL The Kramers-Heisenberg-Waller (KHW) matrix element for the two-photon elastic scattering may be written in the second order of the perturbation theory as: M = M( Ω ) + M( Ω ), () where Ω = ω+ E + i, Ω = ω+ E i, () s s ik r ik r < i e ( α s) n>< n e ( α s) i> M( Ω )= mlim, 0 E (ω + E + i) n n s ik r ik r < i e ( α s) n>< n e ( α s) i> M( Ω )= mlim, 0 E ( ω+ E i) n n s (3)

3 988 K. Karim et al. 3 and ω, k j =ω ν, s j are the energy, momentum and polarisation of the incident ( j = ) and scattered ( j = ) photons. The energy of the initial state of the target electron, which is the same as the final state, meaning the s / Dirac state in coulombian field is Es m + γ = with γ= ( α Z ) /. The sum over the complete set of intermediate n > implyes both negative and positive frequencies contributions. The α s operator is expressed in terms of the Dirac 4x4 matrices. The quantity >0, arbitrary small, allows avoiding the singularities when E =ω+ E or E = ω+ E. (4) n s n s Following the procedure presented in some previous papers [7, 8, 0], we could obtain the nonrelativistic limit of the Rayleigh amplitudes in terms of the invariant amplitudes M = M( ωθ, ) s s + N( ωθ, ) s ν s ν. (5) ( ) ( )( ) The differential cross section of the Rayleigh scattering (per s electron) for unpolarized photons is where r ( A + A ) d = 0, dω (6) ( ) ( ) ( ) A = M ωθ, and A = M ωθ, cos θ N ωθ, sin θ. (7) The amplitudes M and N in the above formulae have been expressed in terms of the photon energy ω and scattering angle θ in the form M( ωθ, )= O ( θ) P( Ω, θ) P( Ω, θ) (8) and N( ωθ, )= Q( Ω, θ ) + Q( Ω, θ ). (9) [ ] In these relationships, the nonrelativistic limit of the various terms involved for the s subshell have specific expressions that we present in the following. Thus, the nonrelativistic atomic form factor for the s subshell was found as sin( γφ) cos( γφ) δ O ( θ ) = cos[( ) ] sin[( ) ], +γ + γ+ φ γ + φ (0) δ γsinφ cosφ where

4 4 Elastic scattering of X-rays and Γ-rays 989 k 4 = k δ = = ω sin θ, φ= arctan δ, () η η αzm and = k k is the momentum transfer between the scattered photon and the electron while η =αzm / is the mean value of the momentum of the s electron. In the case of the s subshell, the amplitudes involved in the previous expression have more complicated expressions than in the case of K-shell electrons, as follows 5 7 λ Ω 4 X F(4 τ; 4, 4;5 τ; x, x) P( Ωθ, )=6 ( ) 5 γ m δ + 8 ω d (4 τ) X F (3 τ;3,3; 4 τ; x, x ) d (3 τ) η m ( γ ) ± Ωω+ α Z Ω α Z m δ + m ω X F( τ;, ;3 τ; x, x) + 4 d ( τ) Ω Ω α Z ω s α Z ( γ ) + ( γ ) ω ω α Z ( γ) ω α Z m Ω α Z Ω + ( ), 3 ± γ + ± + δ 4 ω δ 4 ω 8 ω γ Q( Ωθ, )= 3 ωηm ( γδ ) +γ ( γ X +η) δ 4 ω X F (5 τ;5,5;6 τ; x, x ) d (5 τ) γ + 3 ω η 4+ 3 γ + δ X (7γ + ) + ω 5η +α Z Ω 4ω X d F (4 τ; 4, 4;5 τ; x, x ) 3 5 (4 τ) η ω ( γ ) + α Z 8 3 α Z m m 3 X ± ( γ) α Z ( γ ) + α Z ( γ ) 4 ω 8 ω 8 ω 5 X F(3 τ;3,3; 4 τ; x, x) 3 3 α Z ω ω δ Z, 6 α ± + 4 d (3 τ) 4 4 m m 4 δ + 4 X ω ± () (3)

5 990 K. Karim et al. 5 for each of the cases Ω=Ω or Ω =Ω. In the expressions above ( ) ( ) d = η mω+η X ; d = η mω ηx, while the arguments x and x of the Appell F functions are the roots of the X equation x sx+ u = 0, where u = d / d, s= u and m ω ( ) ( m ) / X Ω = Ω. In these expressions we must consider the nonrelativistic limit of the parameter X as X ( Ω ) = mω+η and also the nonrelativistic limit of the parameter j = Zm / X ( j ) τ α Ω. j Using the optical theorem, the total cross section of the photoect on the s electrons is obtained as 3 τ ( ) π+χ ω ph m Ω 3 m e 3 ωs ωs s = r0 Zef Z 3 ef πτ π α + α 3 ω ω 8 ω e 5 ω m, (4) where is the threshold energy for the s subshell and ω s =( ) m (5) η X η arctan, π + mω + η ω m χ ( ω)= η X η π arctan, > ω mω η In a previous paper [0], we noticed a very good agreement of the results obtained with these formulae, for targets with intermediate and high Z, with other relativistic calculations [, 3] of the total photoect cross-section and angular distribution, if the photon energy is higher than at least one and half the photoect threshold energy. However, for low Z elements and energies closer to the photoect threshold, the limitations of the model based on a constant ective atomic charge lead to increased errors. In order to extend these formulae to low atomic number elements and energies closer to the photoect threshold we propose in this paper some improvements to the previous model. m (6)

6 6 Elastic scattering of X-rays and Γ-rays 99 Considering that the interaction of the incident photon occurs at a distance from the nucleus proportional with the wavelength rint ~ λ ~ (7) ω and taking into account that the dynamic ective atomic charge Z din seen by the photon decreases roughly linearly with the distance, starting from the static value Z st Z = Z C r (8) din st 0 int, we may calculate an ective atomic charge which dynamically varies with the incident photon energy as Z din C = Z. (9) ω st The static value for the ective atomic charge is obtained by fitting the Hartree-Fock wave function with a s coulombian one, as described in [0]. The constant C may be obtained by using an iterative procedure and imposing that the variation at the last step should be less than %. +γ For low Z values the quantity = is close to unity, so that in these cases it is possible to put =. On another hand, the Independent Particle Approximation (IPA) is a poor approach near the photoect threshold, so that we do not expect good predictions in a range just above the threshold. The nonrelativistic limit of the parameter χ ( ) ω that occurs in the exponential in eq. (4) may be written as / η X η E arctan, π + Esω+η ω ( )= η X η π ω Esω η Es χ ω arctan, >, s (0) where E s =m is the energy of the s electron. Using this model we obtained the numerical results for the total photoect cross sections in the nonrelativistic limit shown in Tables 4.

7 99 K. Karim et al. 7 Table The photoect cross-sections (barns) from the s electrons of calcium compared with the results of Scofield [] at various incoming photon energy. The column indicates the differences of the present dynamic screening model, while the column s indicates the differences of the static model in reference [0] s Sco ω s (kev) (barns) (barns) (%) (%) Z s Table The same as in Table, for zinc Sco ω s s s (kev) (barns) (barns) (%) (%) Z Table 3 The same as in Table, for silver s Sco ω s (kev) (barns) (barns) (%) (%) Z s

8 8 Elastic scattering of X-rays and Γ-rays 993 Table 3 (continued) Table 4 The same as in Table, for lead s Sco ω s (kev) (barns) (barns) (%) (%) Z s Concerning the angular distribution of the Rayleigh scattering amplitudes, the dynamic screening model should provide lower values at low energies, especially for lighter elements, compared with the static screening model described in [0]. This may be noticed in Figs. 3.

9 994 K. Karim et al. 9 Fig. The angular distribution of the differential cross section of Rayleigh scattering on the s electrons of calcium for photon energy 6.04 kev. Fig. The angular distribution of the differential cross section of Rayleigh scattering on the s electrons of zinc for photon energy 6.04 kev.

10 0 Elastic scattering of X-rays and Γ-rays 995 Fig. 3 The angular distribution of the differential cross section of Rayleigh scattering on the s electrons of silver for photon energy 0 kev. 3. CONCLUSIONS As it may be noticed from the numerical results in the tables, the presented dynamic screening model considerably improves the predictions for the total photoect cross-section and hence for the angular distribution of the Rayleigh scattering on s electrons in the case of low atomic number elements and energies closer to the photoect threshold. This proves that a coulomb field with an ective charge given by the Hartree-Fock method gives fairly good predictions both for Rayleigh scattering and photoect in a large energy range for any atomic number elements. Acknowledgements. We wish to thank to Prof. A. Costescu for suggesting this work and many helpful discussions. The work was partially supported by the Romanian Executive Unit for Financing Higher Education, Research and Inovation (UEFISCDI), under grant PN II -39/008. The work of C. Stoica was supported by the strategic grant POSDRU/89/.5/S/5885, Project "Postdoctoral programme for training scientific researchers" cofinanced by the European Social Found within the Sectorial Operational Program Human Resources Development

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