Spin dependence in the neutralization of He+ ions in metals: An analysis of different contributions

Nuclear Instruments and Methods in Physics Research B 232 (2005) 8–15 www.elsevier.com/locate/nimb

Spin dependence in the neutralization of He+ ions in metals: An analysis of different contributions M. Alducin

*

Donostia International Physics Center (DIPC), P. Manuel de Lardizabal 4, 20018 San Sebastia´n, Spain Available online 27 April 2005

Abstract We study the spin polarization of the Auger electrons produced during the neutralization of He+ ions in a free electron gas. In this process, one metal electron decays to the unoccupied state and a second electron is promoted to a continuum excited state. Although the spin of the decaying electron is fixed, both spins are allowed for the excited one. The states of the electrons involved in this Auger capture process are described by the spin-dependent Kohn–Sham orbitals obtained from density functional theory and the local spin approximation. The Auger capture rates indicate a strong polarization of the excited electron. In a paramagnetic free electron gas, there are two mechanisms accounting for this effect, the spindependent screening and the interference between indistinguishable processes when the involved electrons are in the same spin state. In a spin-polarized medium, the difference in the density of spin-up and spin-down electrons is a new ingredient to be considered. As a result, the excited electrons preferably come from the majority band, even in the case of He+ ions with spin opposite to that of the majority band embedded in a low spin-polarized free electron gas. Ó 2005 Elsevier B.V. All rights reserved. PACS: 34.50.Dy; 79.20.Rf Keywords: Spin-polarized electron excitation; Polarized He+; Auger neutralization

1. Introduction The interaction of ions/atoms with solids is so complex that a wide variety of phenomena may take place depending on the properties of the pro-

*

Tel./fax: +34 943 018293. E-mail address: [email protected]

jectile and the target. Auger charge exchange and, in particular, Auger capture is an example. In this process, the Coulomb interaction between two electrons in the target induces the decay of one electron to an unoccupied bound state and the promotion of the second electron to an excited continuum state. The relative position of the energy levels and the bandwidth of the target determine the feasibility of this process and the

0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.03.017

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15

energy range of the excited electrons. Since the pioneering work of Hagstrum [1], different experimental techniques have exploited the Auger neutralization of He+ ions with surfaces to study the electronic properties of the target. These techniques bring the possibility of controlling the spin-state of the projectile and measuring that of the electrons emitted during the interation process. Precisely, this feature is what makes them a valuable tool to probe both paramagnetic [2–4] and magnetic materials [5–9]. The interpretation of the spectra obtained in these experiments requires a deep understanding of the whole mechanism taking place. With this aim, different theoretical approaches have been developed in the last years, focussing on different aspects of the Auger process. In most of them, the target considered is paramagnetic [10–14] and only a few analyze the Auger neutralization in magnetic surfaces [15,16]. As shown in [12], one of the main ingredients to treat the problem correctly is the description of the strong perturbation induced by the projectile, particularly, for distances in the proximity of the surface. In this respect, density functional theory (DFT) has been applied to describe different problems in which the density of the total system (projectile and target) plays a crucial role [17,18]. More recently, DFT with the local spin density approximation (LSD) has been used to study the deexcitation of metastable He* atoms at surfaces [19] and the Auger neutralization of He+ ions in paramagnetic materials [20,21]. The latter was extended to the interesting case of a spin-polarized free electron gas (FEG) [22]. The purpose of the present work is to extend the study of the Auger neutralization made in [20–22]. The idea is to analyze and compare the different ingredients that contribute to the spin dependence of the Auger capture (AC) of He+ ions. We start in Section 2 by comparing the role played by the screening and the interference in the paramagnetic case. In Section 3, we analyze how the contribution of those ingredients may change when a spin-polarized FEG is considered. The main results are summarized in Section 4. Atomic units (a.u.) are used throughout this work, unless otherwise stated.

9

2. Auger capture of He+ ions in a paramagnetic free electron gas The AC of a He+ ion is a spin-dependent process even when the ion is embedded in a paramagnetic FEG. In this case, the spin-dependent perturbation is introduced by the He+ ion itself because it has only one electron in the ground state. Here we briefly indicate the theoretical model proposed to study the AC process. A detailed derivation of this model can be found in [20,21]. We assume a He+ ion with a spin-up bound electron (Heþ " ) embedded in a paramagnetic FEG. The latter is characterized by the parameter rs, which is defined from the homogeneous electron density as n0 ¼ 3=ð4pr3s Þ. Since the FEG is paramagnetic, the spin-up (spin-down) electron density induced around a Heþ " ion is totally equivalent to the spin-down (spin-up) electron density induced around a He+ ion with a spin-down bound electron. The spin-dependent electron density is calculated using density functional theory (DFT) and the local spin density (LSD) approximation. The procedure consists in solving self-consistently the Kohn–Sham (KS) equations of the system formed by the Heþ " ion and the paramagnetic FEG. The electron density n(r) is obtained as X X  j 2 ui ðrÞ ; nðrÞ ¼ ð1Þ j¼";# i2occ

where uji ðrÞ are the KS wavefunctions with eigenvalues eji . The electron density for just spin-up (spin-down) electrons n"(r) (n#(r)) can be defined in a similar way by limiting the sum over occupied states to the required spin component. The spin-dependent perturbation introduced by the Heþ " ion is described through the one-electron exchange-correlation (XC) potential that depends on the local density n(r) and on the local spin polarization f(r). The latter is defined as, fðrÞ ¼

n" ðrÞ  n# ðrÞ . n" ðrÞ þ n# ðrÞ

ð2Þ

The results shown here are calculated using the LSD parametrization of [23]. The Heþ " ion is modelled by populating just the spin-up bound KS 1s state of the system.

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15

C" ðHeþ "Þ ¼ 2p

It is worthy to note that this expression can be rewritten as the sum of two terms C# ¼ C#0  Cint . The first one is equivalent to the expression of C", if we just replace the spin-up state of the excited electron by the spin-down. The second one reflects the interference between the two mentioned indistinguishable processes. þ # The AC rate C" ðHeþ " Þ ðC ðHe" ÞÞ is represented in the upper panel of Fig. 1 by thick dash-dotted

10

10

Γj (a.u.)

The AC rate C, i.e. the probability per unit time that an electron in the continuum decays to the unoccupied state bound to the ion whereas a second electron is promoted to an excited continuum state, is calculated in first order perturbation theory. At this point, we use the KS states obtained previously to approximate the initial and final states of the system. Doing so, the AC rate is described as the sum of the probabilities of two different channels: (i) the capture of a spin-down electron and the excitation of a spin-up electron C" ðHeþ " Þ and (ii) the capture and excitation of a spin-down electron C# ðHeþ " Þ. The expression of C" ðHeþ Þ is given by " X X X

u#1 2occ u"2 2occ u"3 62occ

#

C

ðHeþ "Þ ¼ 2p

X X X

u#1 2occ u#2 2occ u#3 62occ

Z 1   

 drdr0 ½u#a ðrÞ ½u#3 ðr0 Þ vðr; r0 Þu#2 ðr0 Þu#1 ðrÞ 2 2 Z   # 0  # 0 # 0 # 0  drdr ½ua ðrÞ ½u3 ðr Þ vðr; r Þu1 ðr Þu2 ðrÞ  

d e#1 þ e#2  e#a  e#3 :

ð4Þ

10-3

10

3

-3

rs (a.u.)

4

5

-4

2

3

4

5

rs (a.u.) 100 LSD LDA&INT

75

ξAC %

where v(r, r 0 ) = 1/jr  r 0 j is the Coulomb potential responsible for the decay. The wavefunctions and the eigenvalues of the electron to be captured are u#1 ðrÞ with e#1 in the initial state and u#a ðrÞ with e#a in the final state. In case of the electron to be excited we use u"2 ðrÞ with e"2 in the initial state and u"3 ðrÞ with e"3 in the final state. The expression for C# ðHeþ " Þ is slightly different. Since the captured and excited electrons have the same spin orientation, there are two indistinguishable processes contributing to C# ðHeþ " Þ: either the initial wavefunction of the captured electron is u#1 ðrÞ and u#2 ðrÞ is the initial wavefunction of the excited electron, or vice versa. As a result one gets,

-2

10-2

2

10

Z 2    " 0  " 0 # 0 # 0 

 drdr ½ua ðrÞ ½u3 ðr Þ vðr; r Þu2 ðr Þu1 ðrÞ  

d e#1 þ e"2  e#a  e"3 ; ð3Þ

10-1

-1 Γ (a.u.)

10

50

LSD0

25

2

3

4

5

rs (a.u.) Fig. 1. Dependence of the AC probability on rs for a Heþ " ion embedded in a paramagnetic FEG. The two channels contributing to the Auger process are represented in the upper panel: C" by dash-dotted curves and C# by dashed curves. The thick ones correspond to the results obtained with the full calculation (LSD). The thick dotted curve shows the results obtained for C# when the interference term is ignored (LSD0). The thin curves (LDA&INT) indicate the results which are calculated using the LDA to parametrize the XC potential. The inset represents the total AC rate C = C" + C# for the three approximations. The lower panel shows the spin polarization of the electrons excited during the AC process: LSD is represented by the thick solid curve, LSD0 by the thick dotted curve and LDA&INT by the thin solid curve.

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15

(thick dashed) line. Interestingly, the comparison þ # between C" ðHeþ " Þ and C ðHe" Þ reveals that the spin-up electrons are preferably excited. The spin polarization of the AC rate, defined as nAC ¼

C"  C# ; C" þ C#

ð5Þ

is shown in the lower panel by a thick solid line labeled as LSD. Our results show a strong spin polarization that increases from 75% to 90% as the FEG electron density decreases. In view of Eqs. (3) and (4), there are two ingredients that may contribute to the spin dependence of the AC process: the spin-dependent screening that accounts for the differences u"i ðrÞ 6¼ u#i ðrÞ and the indistinguishability of electrons with equal spin that gives rise to the interference term Cint. Next, let us analyze the role played by each of these ingredients. The induced density obtained after solving the KS equations, Dn(r) = n(r)  n0, shows that the screening is preferably due to those electrons in the continuum with spin-up (i.e. parallel to that of the bound electron). In particular, an estimation of the charge induced in the continuum for each spin orientation (j = ", #), Z 1 Qjc ¼ 4p drr2 Dnjc ðrÞ; ð6Þ 0

gives that the contribution of Q"c to the screening charge is: 57% for rs = 1; 61% for rs = 2; 66% for rs = 3; 75% for rs = 4; 90% for rs = 5. Note that the screening charge for a He+ ion is Q"c þ Q#c ¼ 1. These values also indicate that the local spin polarization of the induced charge increases as the electron density of the FEG decreases. The spin dependence of the screening is understood in terms of the direct exchange between electrons well separated in energy and space [24], as it is the case for the electron bound to the Heþ " ion and any of the spin-up electrons in the continuum. As a result of direct exchange, the spin-down electrons feel more the Coulomb repulsion that pushes them away from the vicinity of the Heþ " ion. This effect is more relevant at low electron densities, when the contribution of the XC term to the total energy of the system is comparable to the kinetic term that dominates at high densities [25].

11

Turning to the AC process, the above features indicate in agreement with the results of Fig. 1 that first, the screening favours the excitation of spinup electrons, since these are closer to the Heþ " ion and second, the difference between spin-up and spin-down excited electrons increases when the electron density decreases. The excitation of spin-up electrons is also favoured by the interference term that describes the reduced probability of finding two electrons with the same spin, close to each other. Therefore, if a spin-down electron is captured to the unoccupied bound state, the excitation of a spin-up electron is enhanced (or vice versa). Next, the purpose is to compare the contribution of the screening and the interference to the spin polarization of the AC process. The contribution of the screening can be estimated by neglecting Cint in the calculation of C#. The results, which correspond to C#0 , are represented by the thick dotted line in the upper panel of Fig. 1. The corresponding nAC is also indicated by a thick dotted line in the lower panel (LSD0). In order to isolate the contribution of the interference term, the probabilities are also calculated using the KS wavefunctions obtained when the XC potential is represented in the LDA. Within this approximation, the XC potential only depends on the local electron density n(r) and not on the local spin polarization f(r). Therefore, since the KS equations are now spin independent, the resulting continuum induced densities Dn"c and Dn#c are equal. These results are also represented in Fig. 1. In the upper panel, C" ðHeþ " Þ by the thin dash-dotted line and C# ðHeþ ) by the thin dashed line, in " the lower panel nAC by the thin solid line (LDA&INT). A comparison of the three curves shown in the lower panel shows that the interference is the main mechanism accounting for the spin polarization of the excited electrons. When the interference is neglected (LSD0), the probability C# increases around one order of magnitude. As a consequence, the spin polarization nAC(LSD0), which is entirely due to the screening, only varies from 10% to 45%. Notice also the stronger dependence of nAC(LSD0) on the FEG electron density, indicating that the contribution of screening is significant at low densities. On

12

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15

the contrary, comparing nAC(LSD) to nAC(LDA&INT), we observe that the role of the interference is essential and practically independent of n0. The total AC rates C are represented in the inset: C(LSD) by thick solid line, C(LDA&INT) by thin solid line and C(LSD0) by thick dotted line. These are quite insensitive to the model used, despite the differences obtained in the partial rates.

C" ðHeþ #Þ ¼ 2p

X X X

u"1 2occ u"2 2occ u"3 62occ

Z 1   

 drdr0 ½u"a ðrÞ ½u"3 ðr0 Þ vðr; r0 Þu"2 ðr0 Þu"1 ðrÞ 2 2 Z   " 0  " 0 " 0 " 0  drdr ½ua ðrÞ ½u3 ðr Þ vðr; r Þu1 ðr Þu2 ðrÞ  

d e"1 þ e"2  e"a  e"3 ;

3. Auger capture of He+ ions in a spin-polarized free electron gas

for the capture and excitation of spin-up electrons and C# ðHeþ #Þ

Contrary to the paramagnetic case studied in the previous section, in a spin-polarized FEG there is a distinctive spin orientation: that parallel to the spin majority band. Therefore, in order to study the AC of He+ ions embedded in a spinpolarized FEG, two different possibilities have to be considered. The spin of the electron bound to the He+ ion is parallel either to the majority or to the minority band. The theoretical model of Section 2 can be generalized by introducing a few modifications. Now the FEG is characterized by the electron density n0 and the spin polarization defined as  f0 ¼ 

n"0  n#0 n"0 þ n#0

 .

ð7Þ

As explained in Section 2, the electron density is obtained by solving the KS equations in a selfconsistent manner. The XC potential is treated within the LSD approximation. Here, the local spin polarization f(r) is created by the He+ ion and by the FEG. The study of the AC process also needs care. Let us consider that the spin majority band is given by the spin-up electrons. Hence, f0 > 0 in our case. When the spin of the electron bound to the He+ ion is parallel to the majority band (Heþ " ), Eqs. (3) and (4) give the AC rate in which the excited electron has spin-up (C" ðHeþ " Þ) and spin-down (C# ðHeþ Þ), respectively. Similarly, " the expressions of the AC rates for He+ ions with a spin-down bound electron (Heþ # ) are

ð8Þ

¼ 2p

X X X

u"1 2occ u#2 2occ u#3 62occ

Z 2    # 0  # 0 " 0 " 0 

 drdr ½ua ðrÞ ½u3 ðr Þ vðr; r Þu2 ðr Þu1 ðrÞ  

d e"1 þ e#2  e"a  e#3 ; ð9Þ for the capture of a spin-up electron and the excitation of a spin-down electron. In case of Heþ # ions, the corresponding interference term appears in C" ðHeþ # Þ, since both the captured and the excited electrons have the same spin state. Here we focus on the dependence of the AC process on the spin polarization of the FEG also characterized by an electron density parameter rs = 2.12 a.u. This density value together with f0 = 0.27 are typically used to represent a Fe substrate [26]. There are three ingredients accounting for the spin dependence of the AC event: the screening, the interference term and the different number of spin-up and spin-down electrons to participate in the capture process. It is worthy to remark that the latter also affects the behaviour of the screening process itself. Despite the direct exchange favours the alignment of electrons with spin parallel to that of the bound electron, this behaviour is no longer true in case of a Heþ # ion embedded in a highly spin polarized FEG. In such a case, the number of spin-down electrons is insufficient to provide the screening of the external charge, hence the spin-up electrons take the role. The contribution of the majority band to the þ screening of Heþ " and He# ions is shown in Table 1

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15 Table 1 Contribution of Q"c to the screening charge for the two possible spin states of the electron bound to the He+ ion and different values of the FEG spin polarization f0 = 0.27

f0 = 0.5

f0 = 0.9

62% 38%

67% 44%

77% 55%

98% 86%

The FEG electron density corresponds to rs = 2.12.

rs = 2.12 a.u. +

He

0.03

Γ (a.u.)

Q"c (Heþ ") Q"c (Heþ #)

f0 = 0.0

13

↑

0.02

0.01

0

0

0.2

0.4

0.8

0.6

ζ0 rs = 2.12 a.u. +

0.03

Γ (a.u.)

for different values of the FEG spin polarization f0. An interpolation of the data for the Heþ # ion gives that the contribution of Q"c to the screening charge is indeed greater than 50% for a spin polarized FEG with f0 P 0.4. The AC rates for Heþ " ions (upper panel) and Heþ ions (lower panel) are shown in Fig. 2 as a # function of the FEG spin polarization. In each panel, C is represented by solid lines, C" by dashdotted lines, and C# by dashed lines. At f0 = 0 (paramagnetic FEG), both ions are equivalent þ thus CðHeþ " Þ ¼ CðHe# Þ. Furthermore, since the spin perturbation only originates from the electron þ # bound to the He+ ion, C" ðHeþ " Þ  C ðHe# Þ and þ þ # " C ðHe" Þ  C ðHe# Þ as we indeed obtain. Interestingly, the total AC rate decreases as f0 increases for both ions, being steeper for Heþ " . This behaviour can be understood by analysing separately the f0-dependence of C" and C#. First, we focus on the neutralization of Heþ " ions, in which a spin-down electron is captured to the unoccupied KS bound state. The behaviour of C# with f0 is a consequence of reducing the number of spin-down electrons that are the only ones participating in this process. The initial increase and further decrease of C" is explained as a competition between the increasing number of available electrons to be excited (spin-up electrons) and the decreasing number of electrons to be captured. Although the former enhances the effect of the spin-dependent screening, that favours the excitation of spin-up electrons, the latter rapidly dominates and gives rise to the decrease of C" from f0  0.2. Now let us analyze the neutralization of Heþ # ions by capturing a spin-up electron from the conduction band. For f0 = 0, the spin-dependent screening and the interference favour the

He

↓

0.02

0.01

0

0

0.2

0.4

0.6

0.8

ζ0 Fig. 2. Dependence of the AC probability on f0 for a He+ ion embedded in a spin-polarized FEG with rs = 2.12 a.u. In the þ upper (lower) panel, the results are obtained for a Heþ " (He# ) ion. In each panel, dash-dotted (dashed) lines correspond to the excitation of a spin-up (-down) electron C"(C#). The total probability C = C" + C# is shown by solid lines.

excitation of spin-down electrons (C# ðHeþ #Þ > C" ðHeþ Þ). As f increases, we observe a rapid 0 # decrease of C# and a monotonic increase of C". This behaviour is explained as a competition between the interference and screening. As f0 increases, the screening becomes dominated by the spin-up electrons, which are the numerous ones (see Table 1). Therefore, although the interference still favours the excitation of spin-down electrons, the screening facilitates the excitation of spin-up electrons. Furthermore, the reduction (increase) of the number of spin-down (-up) electrons to be excited as f0 increases also contributes to the

14

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15

decrease (increase) of C# (C"). Despite all, only for a highly magnetized FEG (f0 > 0.7), the excitation of spin-up electrons is more probable (C" ðHeþ #Þ > " C# ðHeþ Þ). Nevertheless, the increase of C is not # enough to compensate the rapid reduction of C# and, as a result, we simply observe a smoother decrease in the total probability of neutralizing a Heþ # ion as compared to that of neutralizing a Heþ " ion. The spin polarization of the electrons excited during the capture process (see Eq. (5)) is represented in Fig. 3 for an embedded Heþ " ion (solid line) and Heþ ion (dashed line). The competition # between the spin-up and the spin-down electrons is especially interesting in the latter (Heþ # ions). In this case the spin polarization of the excited electron varies from negative to positive values. For slightly spin-polarized FEG (f0 < 0.27) the interference and still the screening (see Table 1) favour the excitation of spin-down electrons in spite of being less abundant. For highly spinpolarized FEG (f0 > 0.6), the number of spindown electrons is so small that the excitation of spin-up electrons is clearly favoured. At the typical spin polarization of Fe (f0 = 0.27), the neutralization of Heþ ions is # þ slightly more probable than that of He" ions (see

80

+ ↑

4. Summary

He

ζ AC %

40

0

+ ↓

-40

He

-80 0

0.2

Fig. 2). In both cases, the excitation of electrons with spin parallel to that of the bound electron is favoured. More precisely, the polarization of the excited electrons is nAC  83% for Heþ " ions and nAC  53% for Heþ ions. The asymmetry # þ obtained, j nAC ðHeþ Þ j>j n ðHe Þ j, is due to the AC " # FEG spin polarization. Finally, we compare the effect of the screening and the interference in the spin dependence of the AC process. Fig. 3 also shows the spin polarization obtained by neglecting the interference term (Heþ ", by thin solid line) and (Heþ , by thin dashed line). In # case of Heþ " , the spin polarization obtained for f0 6 0.5 is mainly a consequence of the interference term. For higher f0 values, the role of screening, which is clearly dominated by the abundant spinup electrons is also important. In case of Heþ # , the competition between the interference term and the reduced number of spin-down electrons that also determines the screening, is manifested in this figure. When the interference term is neglected, nAC is positive practically in all the f0 range. This shows that, except for the paramagnetic limit, the interference is the only ingredient favouring the excitation of spin-down electrons. Up to f0  0.6 its contribution dominates the sign of the electrons to be excited. However, for highly spin-polarized FEG this sign is reversed.

0.4

0.6

0.8

ζ0 Fig. 3. Spin polarization of the Auger capture rate nAC, defined in Eq. (5), as a function of the FEG spin polarization f0. Thick (thin) lines indicate the results obtained by the LSD (LSD0) when the interference term is (not) included. Solid lines þ correspond to the Heþ " ion and dashed lines to the He# ion. The zero spin polarization is indicated by the straight solid line.

We have studied the spin-dependent Auger neutralization of He+ ions embedded in a free electron gas. When the medium is paramagnetic the spindependent perturbation is created by the ion. The spin polarization of the Auger process is strong, 75–90% for the typical metal electron densities (2 6 rs 6 5). There are two ingredients accounting for this effect. On the one hand, close to the He+ ion, the spin-dependent screening favours the presence of those electrons with spin parallel to that of the electron bound to the He+ ion. This contribution is important at low electron densities. On the other hand, the reduced probability of finding two electrons with the same spin close to each other enhances the emission of a spin-up(-down) electron when a spin-down(-up) electron is captured.

M. Alducin / Nucl. Instr. and Meth. in Phys. Res. B 232 (2005) 8–15

This is the most important ingredient in all the range of metal densities. In a spin-polarized medium, the difference in the density of spin-up and spin-down electrons is a new factor to be considered. When the spin of the bound electron is parallel to the majority band, the screening, the interference and the spin polarization of the medium favour the excitation of electrons from this band. The interference is also here the essential mechanism contributing to the Auger spin polarization. In case of He+ ions with spin opposite to the majority band, the spin polarization of the Auger electrons change more drastically as the spin polarization of the medium is increased. Whereas for low values, the minority spin electrons are preferably excited due to interference, for higher values, most Auger electrons come from the majority band.

Acknowledgements Scientific collaboration with R. Dı´ez Muin˜o and J.I. Juaristi is gratefully acknowledged. I acknowledge financial support by the Gipuzkoako Foru Aldundia. This work has been partially supported by the Basque Departamento de Educacio´n, Universidades e Investigacio´n, the University of the Basque Country UPV/EHU (Grant No. 9/UPV00206.215-13639/2001) and the Spanish MCyT (Grant Nos. BFM2001-0076 and MAT2001-0946).

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