Compton scattering cross section for inner-shell electrons in the relativistic impulse approximation

Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

Contents lists available at ScienceDirect

Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Compton scattering cross section for inner-shell electrons in the relativistic impulse approximation G.E. Stutz ⇑ Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina

a r t i c l e

i n f o

Article history: Received 18 October 2013 Available online 27 November 2013 Keywords: Inelastic X-ray scattering Compton scattering Cross section Inner-shell electrons

a b s t r a c t Total Compton scattering cross sections and inelastic scattering factors for bound electron states of several elements have been evaluated in the framework of the relativistic impulse approximation (RIA). The accuracy of different approximate expressions for the singly differential cross section within the RIA is discussed. Accurate evaluations of bound state scattering factors require the use of the full RIA expression. Compton scattering from K-shell electrons dominates over the photoelectric absorption at higher energies. Energy values at which the Compton interaction become the main process of creation of K-shell vacancies are assessed. The role of binding effects in Compton processes at lower energies are clearly evidenced by the computed total cross sections. Calculated K-shell ionization total cross sections, defined as the sum of the photoelectric absorption and the Compton scattering cross sections, are in good agreement with available experimental data. The total Compton cross section for the 2s atomic orbital exhibits a shoulder-like structure, which can be traced back to the node structure of the 2s wave function. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The inelastic X-ray scattering spectroscopy in the regime of large energy and momentum transfers, the so-called Compton regime, has proved to be a powerful technique to investigate ground state properties of valence electrons in condensed matter [1,2]. In this sense, synchrotron radiation-based Compton experiments with high momentum-space resolution have been applied to Fermiology studies on a large variety of systems, among them simple metals [3], substitutional alloys [4] and high Tc superconductors [5]. Provided that the energy transfer is larger than electron binding energies, Compton events may occur with the tightest electrons, and thus, give rise to vacancies in deep atomic orbitals. In some situations, the vacancy generated can be the most interesting consequence of the scattering event rather than the inelastically scattered photon. Inner-shell vacancies will be then filled by radiative (X-ray fluorescence emission) or non-radiative (Auger transition) processes. This way, Compton scattering by core electrons provides an additional channel for the creation of inner-shell vacancies, besides photoelectric absorption. As Compton scattering is the main interaction process for X-rays of several hundreds of keV in a wide range of atomic numbers, Compton processes involving bound electrons may play an important role in many areas. In

⇑ Tel.: +54 3514334051. E-mail address: [email protected] 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.11.001

radiology and radiation therapy, Compton collisions with K-shell electrons could significantly contribute to the absorbed dose [6]. Calculations of experimental dose limits in macromolecular crystallography showed that Compton scattering should be taken into account for incident energies above 20 keV [7]. In biomolecular imaging, where short radiation pulses from X-ray free electron lasers are planed to be used, the Compton scattering along with the photoelectric absorption should be the primary source of sample damage [8,9]. Several studies suggested a correlation between cell inactivation and inner-shell ionizations of DNA atoms [10–12], where Compton scattering of hard X-rays may be responsible for the cell damage during X-ray irradiation. Experiments concerning Compton scattering from bound electrons have been performed for elements of medium and high atomic numbers using coincidence techniques. Most of the experiments focused on scattering from K-shell electrons and aimed to yield experimental data for the doubly differential cross section or, in a few cases, for the singly differential cross section at some selected scattering angles. These experimental works have been summarized and discussed in several reviews by Kane [13–15]. Theoretical aspects of the Compton scattering from bound electrons have been reviewed by Pratt et al. [16]. Different theoretical approaches have been developed and mainly applied to calculate doubly differential cross sections for K-shell electrons. A new theoretical contribution in this field has been made by Drukarev et al. [17], but this work is focused on low photon energies and provides scattering cross sections in a nonrelativistic treatment and only for hydrogen-like atoms.

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

Studies on the total cross section for Compton scattering by inner-shell electrons are very scarce and are limited to a few experimental works [15]. Reported measured values were interpreted on the basis of the simple Klein–Nishina theory for free electrons. It is the aim of this work to compute total Compton cross sections for inner-shell electrons and to evaluate to which extent Compton processes contribute to inner-shell ionizations as compared to photoelectric absorption. Compton cross sections for 1s-, 2s- and 2p-electrons will be obtained from the doubly differential scattering cross section in the framework of the relativistic impulse approximation (RIA) [18,19]. This theoretical approach allows both electron binding effects and the electron momentum density distribution to be taken into account in a rather simple way. The dependence of the singly differential cross section for K-shell electrons on the incident photon energy, atomic number and scattering angle will also be discussed. 2. Basic relations Several expressions for the doubly differential cross section (DDCS) for Compton scattering were derived in a relativistic treatment by Eisenberger and Reed [18] and Ribberfors [19] under several simplifying assumptions within the limits of the impulse approximation for photon–electron interactions [20]. Ribberfors [19] succeeded in obtaining a simple expression for the relativistic DDCS, which is valid for all scattering angles, factorizing it into a kinematical factor and a scattering system-dependent function (the Compton profile). In this framework, the DDCS for electrons of the nlj atomic level and for an unpolarized incident photon beam can be written as

  2 d rnlj r20 E0 ðmcÞ2 1 0 ¼ 2 E q ½ðcp Þ2 þ ðmc2 Þ2 1=2 dXdE z ðR; R0 Þ Z nlj J nlj ðpz Þ HðE  E0  Enlj Þ;

ð1Þ

where r0 is the classical electron radius, m the electron mass, c the velocity of light and q the magnitude of the scattering vector:

k0  h~ kj ¼ q ¼ jh~

1=2 1 02 ðE þ E2  2EE0 cos hÞ c

ð2Þ

~ k (E0 and h k0 ) are the energy and momentum of the inciE and  h~ dent (scattered) photon. The angle h is the polar scattering angle measured from the incidence direction. pz is the projection of the electron momentum in the initial state on the direction of the scattering vector:

pz ¼

~ q EE0 ð1  cos hÞ  mc2 ðE  E0 Þ p ~ ¼ q c2 q

ð3Þ

In the last expression the relativistic energy of the electron was approximated by its rest energy mc2 [18]. The factor XðR; R0 Þ in Eq. (1) is defined by

   2 R R0 1 1 1 1 XðR; R0 Þ ¼ 0 þ þ 2m2 c4  0 þ m4 c8  0 ; R R R R R R

ð4Þ

where R and R0 are given by

h  i p 2 1=2 R ¼ E ðcpz Þ2 þ ðmc2 Þ þ ðE  E0 cos hÞ z q

ð5Þ

0

0

R ¼ R  EE ð1  cos hÞ

ð6Þ

The function J nlj ðpz Þ is the Compton profile of electrons of the nlj atomic orbital, defined as

J nlj ðpz Þ ¼

ZZ

qnlj ð~ pÞ dpx dpy ;

qnlj ð~ pÞ ¼ jvnlj ð~ pÞj2 being the electron momentum density distribupÞ the momentum-space wave function for the nlj tion and vnlj ð~ atomic orbital. An isotropic electron momentum distribution was assumed in Ref. [19] in the derivation of Eq. (1). Anisotropy and polarization effects were introduced by Ribberfors in Ref. [21]. The normalization condition Z

þ1

J nlj ðpz Þ dpz ¼ 1

ð7Þ

ð8Þ

1

on the orbital Compton profiles is assumed. The step function HðE  E0  Enlj Þ accounts for the possibility of exciting an electron of the nlj orbital only if the transferred energy is larger than its binding energy Enlj . The occupancy of the orbital is considered through Z nlj . The DDCS in the framework of the RIA is a quite realistic description of inelastic scattering processes in the Compton regime since it accounts for electron binding effects and also for Doppler broadening of the Compton line through the Compton profile. This cross section has been widely used in the analysis of experimental Compton spectra [2] since it provides a simple connection between the DDCS and the Compton profile, from which valuable ground state information of valence electrons can be obtained. On the other hand, Eq. (1) is of practical interest because the simple relationship of the DDCS to a target-dependent function allows systematic evaluations of integrated cross sections for any atomic orbital using tabulated values for J nlj ðpz Þ. The validity of the factorization of the DDCS in a relativistic context was investigated by Holm [22]. Deviations of only a few percent around the peak center (pz  0) were found for high scattering angles, except for high Z atoms, for which the deviations could be somewhat higher. These discrepancies diminish for smaller scattering angles. The accuracy of the relativistic impulse approximation was the object of several works [23–26]. Tests were made by comparing RIA cross sections with exact S-matrix calculations in the independent particle approximation [27,28]. Despite same small deviations in the peak region, which is primarily a shift of the spectrum, Pratt et al. [16,25] argued that the RIA should be adequate to perform calculations of doubly differential Compton cross sections for bound electrons. Since this work is concerned with integrated Compton cross sections, those discrepancies should not affect the final results to a large extent. Contributions of the ~ p ~ A interaction term (not considered in the RIA) in the peak region, would be appreciable only for scattering from high Z K-shells [16,25]. In order to compute singly differential and total cross sections for bound electrons under the RIA, some approximated expressions have been proposed. These will be briefly discussed in the next sections. 3. Results 3.1. Singly differential cross section. Scattering factor Singly differential cross sections (SDCS) for Compton scattering by electrons of a given atomic orbital are obtained by integrating the DDCS over the allowed range of scattered energies

drnlj ¼ dX

and

9

Z 0

EEnlj

2

d rnlj 0 0 dE dXdE

ð9Þ

Several simplifying assumptions have been proposed to evaluate singly differential cross sections in the RIA formalism. Since the Compton profile has a maximum at pz ¼ 0 and tends to zero as pz ! 1, only those electrons with values of pz close to zero contribute mostly to the DDCS. Under this assumption, Ribberfors and Berggren [29] made the simplifications pz ¼ 0 and, corre-

10

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

spondingly, E0 ¼ EC ¼ E=ð1 þ Eð1  cos hÞ=mc2 Þ (the Compton energy) in the prefactors of Jðpz Þ in Eq. (1). Changing the integration variable from E0 to pz , they obtained for the SDCS

Z

drnlj drKN ¼ Z nlj dX dX

pmax nlj

J nlj ðpz Þ dpz ;

ð10Þ

1

drKN =dX being the Klein–Nishina differential cross section. 0 pmax nlj ¼ pz ðE ¼ E  Enlj Þ denotes the maximum allowed pz value, which corresponds to the largest transferred energy to electrons bound to the nlj orbital. An additional approximation which consists of extending the lower integration limit from pz ðE0 ¼ 0Þ ¼ mc to 1 was made in Ref. [29]. In order to further simplify the evaluation of the SDCS, those authors computed the integral over Compton profiles in a very simple way using a ‘‘linear approximation’’ [29]. A more accurate approximation, which considers the pz -dependence of the kinematical factors in Eq. (1) to first order, was proposed by Brusa et al. [30]. Analogously as in Ref. [29], they set pz ¼ 0 and E0 ¼ EC , but only in the X-factor of the DDCS, and retained the first-order term of the Taylor series expansion around pz ¼ 0 in the remaining factors of Eq. (1). After doing these approximations they found for the SDCS

Z

drnlj drKN ¼ Z nlj dX dX

pmax nlj



1þ

1

To investigate the validity of the approximations leading to the expressions of Eqs. (10) and (11), the corresponding scattering factors are compared with the exact RIA result for K shell electrons in Fig. 1. It is clear that the linear approximation [29] can be useful for quick evaluations of scattering cross sections for electrons from a given atomic orbital when high accuracy is not claimed. Integrals in Eqs. (10)–(12) were numerically accomplished using theoretical orbital Compton profiles calculated by Biggs et al. [33]. These profiles are based on nonrelativistic Hartree–Fock wavefunctions for 1 6 Z 6 36 and relativistic Dirac–Hartree–Fock wavefunctions for 36 6 Z 6 102 and are tabulated for the elements in the range 0 < pz < 100 au. In order to be able to evaluate the integrals, when the upper integration limit exceeded 100 au, tabulated profiles were analytically extended beyond 100 au using profiles of the functional form [30] 2

J ext ðpz Þ ¼ að1 þ bjpz jÞ exp½ð1 þ bjpz jÞ 

ð13Þ

   cqc EC ðEC  E cos hÞ pz J ðp Þ dpz ; 1þ c2 q2c E mc nlj z ð11Þ

where the integration limits were set as in Ref. [29]. qc is the magnitude of the scattering vector at E0 ¼ EC . This approximate expression works acceptably well for evaluations of DDCS for whole atoms of light elements at intermediate and large scattering angles [30]. It should also be mentioned that this simplified RIA-formula was originally proposed to derive an efficient sampling algorithm for the simulation of Compton interactions in a Monte Carlo code [30]. The SDCS’s in Eqs. (10) and (11) factorize into a photon–electron interaction factor, the Klein–Nishina cross section, and a targetdependent scattering factor, as in the Waller–Hartree model. However, it should be noted that these scattering factors are functions of pmax nlj instead of the variable x ¼ sin h=k (k being the wavelength of the incident photon). The variable x has been commonly used to parameterize the whole atom incoherent scattering function in the Waller–Hartree formalism (see, e.g., [31]). An extensive tabulation of atomic scattering factors as a function of the parameter x, but only for one scattering angle (h ¼ 60 ), was made by Kahane [32] by using Eq. (10) and the linear approximation of Ref. [29] to accomplish the integral over pz . This approximation works acceptably well but only for evaluations of whole atom scattering factors [29,32] and at high x values [32]. In the region x < 1 Å1, the linear approximation fails, giving rise to significant discrepancies [32]. When applied to evaluating atomic orbital scattering factors its accuracy worsens for the tightest bound electrons [29,32]. It is clear that after integrating the whole RIA DDCS over scattered energies, a natural factorization of the SDCS does not occur. Nevertheless, a scattering factor for the atomic orbital nlj can be defined as the ratio of the SDCS to the free-electron Klein–Nishina differential cross section [28,36]

R EEnlj Snlj ðE; hÞ ¼

0

d2 rnlj dXdE0

drKN dX

0

dE

ð12Þ

It should be noted that in the exact RIA, the scattering factor is not a function of a single variable, but it depends separately both on the scattering angle and on the incident photon energy. Scattering factors in the Waller–Hartree formalism or those derived from the approximated expressions (10) and (11) depend on a single parameter, x ¼ sin h=k or pmax nlj , respectively.

Fig. 1. Inelastic scattering factors for 1s electrons of Al (a) and Au (b), as a function of the scattering angle, from different approximations: exact RIA (Eq. (12)) (solid line), Ref. [29] (dashed line) and Ref. [30] (dotted line). Scattering factors are normalized to one electron. The scattering factor from Ref. [29] evaluated using the linear approximation is also shown (dash-dotted line).

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

11

Fig. 2. Inelastic scattering factors from Eq. (12) for 1s electrons of C (a), Al (b), Cu (c), Ag (d) and Au (e), as a function of the scattering angle and for different photon energies. Scattering factors are normalized to one electron.

12

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

The parameters a and b are determined from fulfilling the continuity condition at pz ¼ 100 au and the normalization condition for the whole profile according to Eq. (8). As shown in Fig. 1, Eq. (10) without linear approximation can yield scattering factors quite accurately only in the case of light targets and at high incident energies. This is due to the fact that in this limit the whole Compton peak lies within the range of allowed scattered energies. At lower incident photon energies, for which E  Enlj < EC and so pmax nlj < 0, only bound electrons with pz < 0 are able to participate in the scattering process. In consequence, the approximations E0 ¼ EC and pz ¼ 0 in the X factor are no longer appropriate. Keeping first order terms in pz =mc, as proposed in Ref. [30], significantly improves the accuracy of the scattering factor at low energies. Nevertheless, important deviations of the scattering factors from Eq. (11) are noticeable at high scattering angles and, mainly, for heavy elements. This shows that retaining pz first-order terms is not accurate enough to account for the wide range of excited pz values at nearly backscattering events. It can be distinguished for high atomic numbers that the RIA S1s is nonzero at / ¼ 0. Scattering factors evaluated in the approximations of Eqs. (10) and (11) go also to a nonzero value R mc J ðpz Þdpz as / ! 0, which is independent of E and some1 1s what lower than the RIA value. In the latter case, the non-vanishing scattering factor is a direct consequence of having extended the lower integration limit to 1; otherwise Eqs. (10) and (11) would yield a null value at / ¼ 0. Scattering factors in the exact RIA for K shell electrons of different elements from C to Au are shown in Fig. 2. Electron binding energies were taken from the database by Deslattes et al. [34], with the exception of C, for which values from Ref. [35] were used. At high incident energies, the overall trend of S1s is similar to that of the whole atom [31]. At large scattering angles, it approximates to the unity since almost the whole range of pz values are excited. On the other hand, for diminishing scattering angles, it decreases rapidly due to electron binding effects. For diminishing energies, as the range of kinematically allowed pz values is reduced to a large extent, the scattering factor decreases significantly from unity, even at backscattering. Some anomalous behavior at high energies can be distinguished for the heavier elements. S1s is not a monotonously growing function, but it exhibits a broad maximum in the region of intermediate scattering angles and then decreases to values slightly lower than unity at backscattering. Such phenomenon can be observed for Ag and Au at 1000 keV. Similar behaviors were

observed in scattering factors calculated in the framework of the S-matrix theory [28,36]. Experimental evidence for this effect in the K-shell scattering factor was found by Wolff et al. [37,38] in Pb and Sn. Near the upper integration limit, which is set by the H-function in Eq. (1), the energy transfer is close to the electron binding energy and, therefore, the assumptions of the impulse approximation should no longer be valid. At a given energy E, for large scattering angles, for which EC ðhÞ  E  Enlj , the maximum of the Compton spectrum is well inside the range of excited pz -values. In this case, since the main contribution to the integral arises from scattered energies close to the Compton energy, potential deviations of the RIA in the description of the DDCS at E0  E  Enlj should have a negligible effect on the SDCS. On the other hand, in the limit of small scattering angles, for which the Compton energy is outside the integration interval (E  Enlj < EC ðhÞ), scattered energies close to E  Enlj provide the major contribution to the differential cross section. In this case, the SDCS may be affected by deviations of Eq. (1) due to the non-validity of the RIA in the region E0 K E  Enlj . Hence, for a given incident photon energy, a critical scattering angle h , below which some deviations of the RIA-SDCS might be expected, could be defined from EC ðh Þ ¼ E  Enlj , in accordance with the condition stated by Kane (Eq. (12) in Ref. [14]). Nevertheless, it should be noted that h does not represent a sharp threshold for the validity of the impulse approximation. A more realistic condition appears to be weaker than the condition above, as pointed out by Kane [14]. Indeed, Laukkanen et al. [40] showed that the RIA formalism describes fairly good the experimental Ag K-shell DDCS at scattered energies close to E  E1s even at intermediate momentum transfers. h for different photon energies is shown in Fig. (3) for the case of Cu. An alternative criterion for the validity of the impulse approximation was formulated by Suric´ [23] in terms of the average electron momentum contributing to the Compton scattering spectrum, but this condition seems to be stronger than necessary [14]. At energy transfers close to E  Enlj the scattering regime for non-resonant Raman scattering of X-rays is met. A non-relativistic theoretical description of this scattering process was given by Mizuno and Ohmura [39] in the one-electron approximation. In this regime, solid state effects, as those introduced by the density of unoccupied electronic states, would be noticeable above the energy-transfer threshold E  Enlj in the energy-loss spectrum [1]. Contributions of resonant inelastic X-ray scattering arising from

Fig. 3. Singly differential Compton cross section for 1s electrons of Cu for different photon energies. The critical angle h (see text) is indicated for each energy by solid points on the corresponding curve.

Fig. 4. Total Compton cross section for the K-shell of C, Al, Cu, Ag and Au in the RIA (solid line). The Klein–Nishina cross section (dashed line) was multiplied by two to account for the occupancy of the K-shell.

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

second order terms can be disregarded, as long as incident energies are far away from electron binding energies. Calculations of the DDCS for bound electrons based on theoretical formalisms beyond RIA were made by different authors [27,28,36]. They used second order S-matrix theory along with self-consistent screened potentials in the independent particle approximation [27,28] or point-Coulomb potentials [36]. The main feature exhibited by those results is the infrared divergence in the soft photon limit, which prevents the DDCS from integrating over the whole scattered energy range. To obtain finite values for the SDCS and, therefore, to give physical meaning to the cross section, low energy cutoffs to the integration were necessary to be assumed. S-matrix results for the K-shell SDCS are strongly dependent on the selected value for the cutoff [28]. Holm and Ribberfors [41] calculated corrections to the nonrelativistic DDCS within the framework of the impulse approximation from an operator expansion. They found that the first correction is asymmetric with respect to pz . Based on this formalism, the Compton profile asymmetry of Na 1s, 2s and 2p electrons was evaluated to be only a few percent [42]. As in this work we are interested in the singly and doubly integrated

13

DDCS, this asymmetry should have no significant effects on the final results. 3.2. Total cross section The total Compton scattering cross section for the nlj atomic orbital is obtained by integrating the DDCS over scattered photon energies and all scattering directions:

rnlj ¼

Z 4p

Z 0

EEnlj

2

d rnlj 0 0 dE dX dXdE

ð14Þ

Because of the isotropy of atomic Compton profiles and of the assumption of a non-polarized incident photon beam, azimuthal symmetry exists. The SDCS has a maximum, whose position moves from h  180 , at low incident energies, to small angles at high incident energies. This is illustrated in Fig. 3 for the case of Cu. It can be seen that the critical angle h is smaller than the scattering angle at which the SDCS reaches its maximum. Therefore, since in most cases the range of scattering angles 0 < h < h contributes negligibly to the whole integral, it is expected that the integrated cross

Fig. 5. Total Compton cross section for the L1 (dotted line), L2 (dashed line) and L3 (short-dashed line) subshell of Al (a), Cu (b), Ag (c) and Au (d) in the RIA. Total L-shell Compton cross section (solid line) is also shown. The Klein–Nishina cross section (dash-dotted line) was multiplied by 8 to account for the occupancy of the L-shell.

14

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

Fig. 6. Theoretical Compton profiles from Ref. [33] for 2s electrons of Al, Cu, Ag and Au.

section be not affected to a large extent by potential fails of the impulse approximation for h < h . In analogy to h , a critical value for the incident photon energy could be defined from EC ðE ¼ E ; h ¼ pÞ ¼ E  Enlj . At incident energies below E failures of the impulse approximation might be expected in all the range of scattering angles. Values of E for the K-shell of C, Al, Cu, Ag and Au are 8.66 keV, 20.76 keV, 52.60 keV, 94.50 keV and 189.54 keV, respectively. For elements of intermediate atomic number, these values are in rough accordance with the minimum energy for the validity of the impulse approximation from the empirical criterion E P 6Enlj by Kane [14]. The E -criterion is more restrictive at low atomic numbers, but more relaxed at high Z, than the criterion from [14]. As commented above for the case of h ; E should not be considered as a sharp energy threshold for the validity of the impulse approximation. An approximated expression for computing total cross sections was proposed by Ribberfors [43]. Starting from the RIA-DDCS and making several simplifying assumptions, a simple algebraic expression for r was derived, which was found to provide acceptable values when applied to evaluating atomic total cross sections. However, since most approximations made in Ref. [43] are based on setting pz ¼ 0 in the DDCS, this simple expression could find applicability only for scattering events from weakly bound electrons and for very high transferred energies. This constitutes a severe limitation if one is interested in evaluating total cross sections for electrons from deeply lying atomic orbitals. Total cross sections for Compton scattering from 1s-, 2s-, 2p1=2 - and 2p3=2 -electrons were evaluated from Eq. (14) for several elements. The K-shell Compton cross section as a function of the incident photon energy is displayed in Fig. 4. In order to make the lower energy behavior clearly visible, cross sections are shown beyond the critical energies. For the sake of comparison, the Klein– Nishina (KN) total cross section is also shown in the same figure. In the limit of high energies, the cross section is nearly independent of the atomic number and approaches the KN-cross section. At large E, since the energy transfer is much larger than binding energies,1 the scattering process can be well described by the KN-theory, which assumes scattering by free electrons at rest. For the case of Au, r1s is slightly lower than the KN-cross section in the limit of very

1 In a strict sense, this is not fulfilled at arbitrarily small scattering angles. The main spectral weight of the DDCS may occur around E0 E  Enlj for small h. Nevertheless, the contribution of dr=dX to the total cross sections is negligible at h  0 since dr=dX decreases rapidly as h ! 0.

Fig. 7. K-shell Compton scattering (solid line) and photoelectric (dashed line) cross section for C (a), Al (b), Cu (c), Ag (d) and Au (e). The total ionization cross section is also shown (dotted line).

15

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

high energies. This is to be attributed to pz -values that are not able to be kinematically excited (1 < pz < mc). 1s Compton profiles of heavy elements spread appreciably over a wide range of electron momentum, even beyond the lower integration limit pz ðE0 ¼ 0Þ ¼ mc ¼ 137:036 au. In the limit of low incident energies, the RIA-cross section vanishes due to binding effects, whereas the KN-cross section goes to its maximum value (the Thomson total cross section) as E ! 0. This marked departure reflects the significance of binding effects on the Compton cross section for bound electrons and consequently the inadequateness of the KN-theory to describe Compton processes in a realistic way at intermediate and low photon energies, even for light targets. At a given incident energy, the total cross section diminishes as the atomic number increases. This behavior is due to the reduction of the energy integration interval for growing Z along with the dependence of J 1s ðpz Þ on Z. Compton scattering total cross sections for each subshell of the L-shell are shown in Fig. 5. Since at large energies binding effects become negligible, the ratio between subshell-cross sections depends only on the occupancy of each orbital. As a result, rL3 is twice that for the L2 - or L1 -subshell in the high energy limit. In the low energy region, a broad shoulder can be observed in the total cross section for the L1 subshell. This structure is to trace back to the nodal behavior of the radial wave function of 2s electrons. This node gives rise to a node in the momentum space wave function and, subsequently, in the corresponding electron momentum density. It can be easily shown that for an isotropic electron momentum density, this quantity is proportional to the first derivative of the Compton profile. Thus, a node in the momentum-space wave function will give rise to a feature in J 2s ðpz Þ with vanishing derivative. This flat region in J 2s ðpz Þ is clearly revealed by the Compton profiles from [33] (see Fig. 6). Assuming 2s hydrogenic wave functions, the node is located at r ¼ 2=Z, where atomic units have been used, and, correspondingly, the wave function in momentum space has a node at p ¼ Z=2 (6.5 au, 14.5 au, 23.5 au and 39.5 au, for Al, Cu, Ag and Au, respectively). As can be seen in Fig. 6, the shoulders appearing in the 2s Compton profiles are around the values predicted by this simple model. After integrating over scattering angles and scattered energies, that well localized structure in momentum space smears out in the energy domain and gives rise to a broad shoulder in r2s . An early analysis of the nodal behavior of hydrogenic wave functions and its effect on the bound-state Compton profile was made by Bloch and Mendelsohn [44]. They predicted shoulder- and dip-like structures in the non-relativistic DDCS for 2s atomic orbitals. In the field of photoionization, a minimum in partial cross sections for atomic targets is induced by the nodes of the radial wave function of the initial state. Such features in the photoionization cross section are usually referred to Seaton–Cooper minima [45]. In this context, the shoulder observed for the 2s atomic orbital could be seen as Seaton– Cooper-like feature in the Compton scattering cross section. To inspect to which extent Compton scattering processes contribute to the generation of atomic inner-shell vacancies, the K-shell Compton scattering cross section is displayed along with

the corresponding photoelectric cross section in Fig. 7. Theoretical values of photoelectric cross sections calculated by Scofield [46] have been used. The ionization total cross section for the K-shell, defined as the sum of the Compton and photoelectric cross sections, is also shown in the same figure. The photoelectric absorption clearly dominates over Compton scattering in the low energy region since the photoelectric cross section goes to its maximum value as E approaches the electron binding energy, the K-shell Compton cross section vanishes rapidly due to binding effects. The situation is reversed at high photon energies because while the photoelectric cross section for a specific atomic shell diminishes monotonously for increasing energy, the Compton cross section reaches a smooth maximum and then decreases slowly. The crossing point energy, at which the ionization is equally probable either by photoelectric absorption or by Compton scattering, depends on the atomic number. K-shell crossing points are located at 31.0 keV, 96.4 keV, 356.6 keV and 1061.7 keV for C, Al, Cu and, Ag, respectively. For the case of Au, the crossing point is outside the energy range of Fig. 7. A method for measuring integrated Compton cross section of inner-shell electrons has been proposed by Verma et al. [47]. Briefly, it consists of measuring the production of fluorescent X-rays emitted by a specific target when it is irradiated with photons of a given energy. The X-ray production is normalized to the X-ray fluorescent intensity at a reference energy for which the contribution of Compton scattering to inner-shell ionizations is negligible small. This method allowed the authors to obtain cross sections with an experimental accuracy of the order of 10%. The experimental results were interpreted in [47] in terms of the free electron model and the departures of the measured data from the Klein–Nishina cross section were investigated. Calculated cross sections for Kshell ionization are compared with experimental values from Ref. [47] in Table 1. Values for the photoelectric cross section at the

Fig. 8. Deviation of the integrated Klein–Nishina cross section from the RIA Compton cross section, relative to the ionization total cross section, for the K-shell of C, Al, Cu, Ag and Au.

Table 1 K-shell cross sections for Compton scattering (rK ), photoelectric absorption (rphoto ) and total (rK þ rphoto ). Cross sections are given in b/atom. Element

Energy (keV)

rK (This work)

rphoto [46]

rK þ rphoto

Experiment [47]

Cu Ag Ag Ag Ag Au Au

279 145 279 662 1250 662 1250

0.640 0.334 0.511 0.465 0.357 0.377 0.321

1.239 67.205 10.302 1.087 0.276 10.490 2.755

1.879 67.539 10.813 1.552 0.633 10.867 3.076

1.95 ± 0.13 75.72 ± 6.36 10.96 ± 0.73 1.50 ± 0.13 0.66 ± 0.04 11.34 ± 0.54 3.16 ± 0.15

16

G.E. Stutz / Nuclear Instruments and Methods in Physics Research B 319 (2014) 8–16

experimental energies were obtained from a spline interpolation of Scofield’s [46] data. The agreement is very good within the experimental uncertainties. Notice that for Ag at 145 keV and 279 keV and for Au at 662 keV the reported experimental uncertainty is larger than the K-shell Compton cross section, so that a reliable validation of the calculated rK by the experiment is not possible in these cases. In order to design such experiments, Fig. 8 can be helpful. This figure displays the energy-dependent deviation of the Klein–Nishina theory from the RIA cross section for the K-shell electrons. This deviation has been normalized to the ionization total cross section, so that it can be taken as a parameter that measures the efficiency of an experiment. At lower energies, though the Compton cross section deviates significantly and progressively from the Klein–Nishina cross section as the photon energy decreases, photoelectric interactions dominate and, consequently, the experiment efficiency would be too low. On the high energy side, since the Klein–Nishina theory succeeds in describing Compton scattering, the deviation diminishes. In the intermediate energy region, the relative discrepancy between both models has its maximum amplitude. Depending on the atomic number, it varies between about 1% and 10% of the ionization total cross section. As a general trend, such maximum occurs at an energy close to the crossing point energy, with the exception of heavy targets, for which it shifts to somewhat higher energies. Irradiation with photons of those energies seems to be the most suitable situation to perform experiments aimed to measure K-shell ionization total cross sections and to investigate deviations from the free electron model. The magnitude of the deviations revealed by Fig. 8 imposes the lowering of the experimental uncertainty to the 1% level. 4. Conclusions Scattering factors and integrated Compton cross sections have been computed on the basis of the relativistic impulse approximation for bounds electrons of several elements ranging from C to Au. Scattering factors evaluated from approximated expressions in the framework of the RIA exhibit deviations which can be significant in some cases. Accurate evaluations of scattering factors for K-shell electrons require using the full RIA Compton cross section. In order to investigate the accuracy of the RIA for computations of scattering factors beyond the critical angle, accurate measurements of the SDCS as a function of the scattering angle would be necessary. RIA total Compton cross sections for bound electrons reflect the significance of binding effects at lower energies as compared to the free electron model. Compton scattering by inner shell electrons dominates over photoelectric absorption at higher energies. The energy at which the probability of creation of K-shell vacancies by Compton interactions compares to that by photoelectric absorption depends on the atomic number of the target, ranging from 31.0 keV for C to 1061.7 keV for Ag. Present results suggest that in order to enhance departures from the free electron model, measurements of K-shell total Compton cross sections should be performed around these energies. Accurate experiments in this direction would be greatly desirable. Integrated Compton cross section for the 2s atomic orbital exhibits shoulder-like structures, which are related to the node in the wave function. Those structures could be experimentally investigated by measuring the total cross section as a function of the photon energy using methods similar to that proposed in Ref. [47]. High flux X-ray beams, tuneable in the energy range of interest, are available at modern synchrotron X-ray sources. The main drawback of these experiments would be its poor efficiency; in addition, the measurements would be highly demanding on statistical accuracy in order to resolve structures in the total cross section. The relativistic impulse approximation provides a simple relationship of the DDCS to the electron momentum distribution for

bound states. It allows systematic evaluations of scattering factors and Compton total cross sections for inner-shell electrons throughout the periodic table. Unlike scattering factors in the Waller–Hartree formalism, tabulations for two input variables (scattering angle and photon energy) will be necessary for RIA scattering factors. Acknowledgment Financial support from SeCyT (Universidad Nacional de Córdoba, Argentina) is gratefully acknowledged. References [1] W. Schülke, Electron Dynamics by Inelastic X-ray Scattering, Oxford University Press, 2007. [2] M.J. Cooper, P.E. Mijnarends, N. Shiotani, N. Sakai, A. Bansil (Eds.), X-ray Compton Scattering, Oxford University Press, 2004. [3] W. Schülke, G. Stutz, F. Wohlert, A. Kaprolat, Phys. Rev. B 54 (1996) 14381. [4] G. Stutz, F. Wohlert, A. Kaprolat, W. Schülke, Y. Sakurai, Y. Tanaka, M. Ito, H. Kawata, N. Shiotani, S. Kaprzyk, A. Bansil, Phys. Rev. B 60 (1999) 7099. [5] C. Utfeld, J. Laverock, T.D. Haynes, S.B. Dugdale, J.A. Duffy, M.W. Butchers, J.W. Taylor, S.R. Giblin, J.G. Analytis, J.-H. Chu, I.R. Fisher, M. Itou, Y. Sakurai, Phys. Rev. B 81 (2010) 064509. [6] D. Terribilini, M.K. Fix, D. Frei, W. Volken, P. Manser, Med. Phys. 37 (2010) 5218. [7] K.S. Paithankar, E.F. Garman, Acta Cryst. D 66 (2010) 381. [8] R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, J. Hajdu, Nature 406 (2000) 752–757. [9] T. Kai, Phys. Rev. A 81 (2010) 023201. [10] A. Boissière, C. Champion, A. Touati, M. Hervé Du Penhoat, L. Sabatier, A. Chatterjee, A. Chetioui, Radiat. Res. 167 (4) (2007) 493. [11] B. Fayard, A. Touati, F. Abel, M.A. Herve du Penhoat, I. Despiney-Bailly, F. Gobert, M. Ricoul, A. L’Hoir, M.F. Politis, M.A. Hill, D.L. Stevens, L. Sabatier, E. Sage, D.T. Goodhead, A. Chetioui, Radiat. Res. 157 (2002) 128. [12] A. Chetioui, I. Despiney, L. Guiraud, L. Adoui, L. Sabatier, B. Dutrillaux, Int. J. Radiat. Biol. 65 (1994) 511. [13] P.P. Kane, Rad. Phys. Chem. 75 (2006) 2195–2205. [14] P.P. Kane, Rad. Phys. Chem. 50 (1997) 31–62. [15] P.P. Kane, Phys. Rep. 218 (1992) 67–139. [16] R.H. Pratt, L.A. LaJohn, V. Florescu, T. Suric´, B.K. Chatterjee, S.C. Roy, Rad. Phys. Chem. 79 (2010) 124–131. [17] E.G. Drukarev, A.I. Mikhailov, I.A. Mikhailov, Phys. Rev. A 82 (2010) 023404. [18] P. Eisenberger, W.A. Reed, Phys. Rev. B 9 (1974) 3237. [19] R. Ribberfors, Phys. Rev. B 12 (1975) 2067. [20] P. Eisenberger, P.M. Platzman, Phys. Rev. A 2 (1970) 415. [21] R. Ribberfors, Phys. Rev. B 12 (1975) 3136. [22] P. Holm, Phys. Rev. A 37 (1988) 3706–3719. [23] T. Suric´, Nucl. Instr. Meth. A 31 (1992) 240. [24] B.K. Chatterjee, S.C. Roy, T. Suric´, L.A. LaJohn, R.H. Pratt, Nucl. Instr. Meth. A 580 (2007) 22. [25] R.H. Pratt, L. LaJohn, T. Suric´, B.K. Chatterjee, S.C. Roy, Nucl. Instr. Meth. B 261 (2007) 175. [26] L.A. LaJohn, Phys. Rev. A 81 (2010) 043404. [27] T. Suric´, P.M. Bergstrom, K. Pisk, R.H. Pratt, Phys. Rev. Lett. 67 (1991) 189. [28] P.M. Bergstrom, T. Suric´, K. Pisk, R.H. Pratt, Phys. Rev. A 48 (1993) 1134. [29] R. Ribberfors, K.-F. Berggren, Phys. Rev. A 26 (1982) 3325. [30] D. Brusa, G. Stutz, J.A. Riveros, J.M. Fernández-Varea, F. Salvat, Nucl. Instr. Meth. A 379 (1996) 167. [31] J.H. Hubbell, Wm.J. Veigele, E.A. Briggs, R.T. Brown, D.T. Cromer, R.J. Howerton, J. Phys. Chem. Ref. Data 4 (1975) 471. [32] S. Kahane, At. Data Nucl. Data Tables 68 (1998) 323–347. [33] F. Biggs, L.B. Mendelsohn, J.B. Mann, At. Data Nucl. Data Tables 16 (1975) 201. [34] R.D. Deslattes, E.G. Kessler Jr., P. Indelicato, L. de Billy, E. Lindroth, J. Anton, Rev. Mod. Phys. 75 (2003) 35. [35] A. Thompson et al., X-Ray Data Booklet, 2009, URL: . [36] I.B. Whittingham, Aust. J. Phys. 34 (1981) 163. [37] W. Wolff, H.E. Wolf, L.F.S. Coelho, S. de Barros, Phys. Rev. A 40 (1989) 4378. [38] W. Wolff, H.E. Wolf, L.F.S. Coelho, Phys. Rev. A 43 (1991) 3543. [39] Y. Mizuno, Y. Ohmura, J. Phys. Soc. Jpn. 22 (1967) 445. [40] J. Laukkanen, K. Hämäläinen, S. Manninen, V. Honkimäki, Nucl. Instr. Meth. A 416 (1998) 475. [41] P. Holm, R. Ribberfors, Phys. Rev. A 40 (1989) 6251. [42] S. Huotari, K. Hämäläinen, S. Manninen, A. Issolah, M. Marangolo, J. Phys. Chem. Solids 62 (2011) 2205. [43] R. Ribberfors, Phys. Rev. A 27 (1983) 3061. [44] B.J. Bloch, L.B. Mendelsohn, Phys. Rev. A 9 (1974) 129. [45] J.-P. Connerade, Highly Excited Atoms, Cambridge University Press, 1998. [46] J. Scofield, Lawrence Livermore Laboratory, Report UCRL-51326, 1973. [47] S.L. Verma, K.L. Allawadhi, B.S. Sood, Nuovo Cimento 63A (1981) 327.