Probing scattering wave functions with nucleus-nucleus bremsstrahlung

NUCLEAR PHYSICS A

Nuclear Physics AS50 (1992) 250-262 North-Holland

Probing scattering wave functions with nucleus-nucleus bremsstrahlung D. Baye, P. Descouvemont' and M. Kruglanski2

Pbvsique Nucléaire 77léorique et Physique Mathématique, C.P. 229, Université Libre de Bruxelles, 8.1050 Brussels, Belgium Received 26 June 1992 Abstract: Nucleus-nucleus bremsstrahlung is explored in the a+a system with four different models : (i) a ten-channel microscopic model with monopolar distortion of the a-clusters, (ii) the elasticcharnel restriction of this microscopic model, (iii) a deep-potential model, and (iv) an equivalent shallow-potential model derived iront supersymmetry . The a + a phase shifts in the different models are very close to each other and to experiment . Different centre-of-mass and laboratory bremsstrahlung cross sections are calculated with an accurate treatment of Coulomb convergence . The single-channel results are close to those of the multichannel calculation . The shallow potential provides rather different cross sections from the other models. Experimental conditions exist where one should easily discriminate between the deep-potential and microscopic models. A confirmation of the microscopic cross sections under these conditions would provide evidence for antisymmetrization effects in scattering wave functions.

1 . Introduction Nucleus-nucleus bremsstrahlung offers a unique possibility of probing the inner part o£ scattering wave functions. Starting with the potential-model description of nucleon-nucleus bremsstrahlung by Philpott and Halderson' ), ab initio calculations have been developed in microscopic' -s), semi-microscopic '1) or potential models'°-"). In principle, experiment should provide away of testing these models and choosing the most efficient ones. This idea has been explored in several works. . '°), deep and shallow nucleus-nucleus potentials are confronted while the In ref potential model is compared with a semi-microscopic model in ref. 9). An approximation to the microscopic model is tested with exact microscopic results in refs. 7"'). 'e-'s) . They However, experimental data are very scarce, with rather large error bars are restricted to a peculiar geometry ofthe detectors, known as the Harvard geometry. All the models agree reasonably well with these data provided they correctly reproduce the phase shifts . Further experimental data are necessary to encourage progress in theoretical descriptions . However, one question arises : which type of Correspondence to: Dr. D. Baye, Physique Nucléaire Théorique et Physique Mathématique, Campus de la Plaine, ULB, C.P. 229, B-1050 Brussels, Belgium. ' Chercheur qualifié FNRS. Z Boursier IRSIA. 0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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data, if any, might help discriminating between models, and with which accuracy. In the present paper, we attempt to answer these questions by performing a detailed comparison between different models for a+a scattering . With tire resonating-group method (RGM), an accurate microscopic description of elastic scattering of light ions is available for many years ",2°). With the help of 2' the generator-coordinate method (GCM) ), a unified description of many different applicatians becomes possible 22). In the GCM, matrix elements of electromagnetic multipole operators are calculated with fully antisymmetric wave functions in a systematic way. The wave functions are derived from a Schrödinger equation involving an effective nucleon-nucleon interaction and the exact Coulomb force. Their sce ".tering asymptotic behaviour is obtained with the help of the microscopic R-matrix method 23.22 ). Recently, improved a+a wave functions involving monopolar distortion of the a-clusters have been derived within the GCM 2°). They allow an accurate reproduction. of the experimental phase shifts. The a +a phase shifts are also very well reproduced by a simple gaussian nucleus-nucleus potential depending on only two parameters 25). Indirectly, this potential is also a success of the microscopic model since its validity relies on the fact that it simulates by deeply bound states, the forbidden states that occur in the microscopic model when the clusters have the same oscillator parameter ") . When the clusters are distorted, the forbidden states are transformed into narrow high25). energy resonances 2°) which still explain the validity of the deep potential of ref. The a+a phase shifts can also be reproduced by more complicated (/-dependent) shallow ,potentials 26). In fact, one can prove that an 1-dependent shallow potential 27). The equivalent exists which is exactly phase equivalent to the deep potential potential s obtained by successive supersymmetry transformations [a simplified 28 approach is presented in ref. )] . It is however singular near the origin in order to verify the correct Levinson theorem 29). Several a + a scattering wave functions are available for studying bremsstrahlung : they can be obtained in the two potential models which provide exactly the same phase shifts and in different types of microscopic models which reproduce these phase shifts to a good accuracy. Therefore, any difference in the bremsstrahlung cross sections will be a measure of differences in the inner part of the wave functions . We shall try to find experimental conditions where these differences are enhanced, as much as possible . When new experiments will be available, they will allow to discriminate between deep and shallow potentials and between microscopic and non-microscopic models . The potential-model description of bremsstrahlung that we employ is detailed in ref.'s) . Special attention is paid to the different convergence problems caused by recall the Coulomb interaction. With the techniques of ref.'s), which we do not ref. 2). here, accurate results are obtained . The microscopic model is described in However, the expressions in ref. 2) are not well adapted to the Coulomb corrections of ref.' s ) and need be reformulated. Also, the GCM in ref. 2) assumes undistorted

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cluster wave functions with a common oscillator parameter. Here, we employ the distorted wave functions of ref. -°) which require a delicate projection on the centre-of-mass (c.m.) motion . Hence the calculation of the matrix elements of the quadrupole operator must be generalized. In sect . 2, the microscopic calculation of bremsstrahlung cross sections is summarized and generalized to distorted wave functions. Numerical results obtained with the different models are compared and discussed in sect. 3. Concluding remarks are presented in sect. 4. 2. GCM description of bremsstrahlung 2.1. CROSS SECTIONS

Two a-particles collide with the relative momentum p; in the z-direction and energy E; = p 21 /2g, where 1c is their reduced mass. After emission of a photon with energy E,, and momentum p,, = hk, in the direction n,, = (®, (p,,) the final momentum of the system is pf in the direction dl, = (®,, (pf). The final energy E1=Pi/21L satisfies Er =Ei -E,,

(1)

up to small recoil corrections. In the a+a system, only even-1 partial waves W1 " are allowed . Hence odd-parity multipoles are forbidden. Since M1 transitions are also forbidden in the longwavelength approximation, the E2 contribution dominates and the c.m. differential cross section reads 15 ) d E,, dfr df,, with

(2 ah) fic  .

(47r)"E'(2 j +1)-' 1 - (221A,-1r.'ÎJIL-li')(221-1Jj0)Y ;"-' (,fl,,),

(3)

where the prime on the summation symbol indicates that only even values are allowed. The matrix element u ;. reads u p=-(1s)'i- Irk;.E'(21f+1)-' / -(1;20AIIfM-) 1.11

x exp[2i(o,,, +S,f)](1p1,J JAz J J p'')Yi ,(Hr)' (4) where 4Q , is an electric quadrupole operator and a, and S1 are, respectively, the Coulomb and quasinuclear phase shifts. In the coplanar Harvard geometry, the a-particles are detected in the directions .fl, = (01 , 0) and 12, = (0=, 0) and the photon remains undetected . The photon energy and final angle are given by E,, = E;[1-(sin 2 0 1 + sinz 0,)/sin 2(01 + 0Z )] , cot 01 =Z(cot 0 1 -cot 0z ),

(5) (6)

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with (pr=0. The Harvard E2 cross section reads

E)2-f

Re (U2.U22* ) 1 xf2(uE,) - + 4(u2 )2+(u where misprints in eq. (60) of ref. '5) are corrected. The cross sections rely on the calculation of U2, . The slowness of convergence of the Coulomb part in eq. (4) requires a special treatment which is detailed in ref' s ). In the following, we only consider the calculation of (IY 'flI"ZfC (7)

2.2. WAVE FUNCTIONS

The construction of a+a wave functions with monopolar distortion of the a-clusters is described in details in ref. `°). Only a brief summary is given here. In partial wave l, the wave function is expanded as V, In, m=

where oii represents the level of monopolar excitation of cluster i. The non-orthogonal wave functions read in RGM notations where .i is the antisymmetrization projector, p= (p, 12p ) is the relative coordinate between the clusters and tA ;l represents an internal wave function of cluster i in state (oi . The internal wave functions are obtained by linearly combining translationinvariant antisymmetric basis functions defined in the harmonic-oscillator model with different oscillator parameters 19 .20) . Therefore, the wave function (8) does not contain any spurious c.m. component. In the GCM, the functions (9) are expanded as a finite linear combination of projected Slater determinants defined in the two-centre harmonic-oscillator model. The distance R between the centres plays the role of a generator coordinate . The restoration of translation, rotation and reflection symmetries requires several steps before providing appropriate basis functions for an expansion of eq. (9). The first 30 step removes the c.m. dependence with a traditional Peierls-Yoccoz projection ) on zero momentum . A second step uniformizes the widths of the relative parts of the translation-invariant basis functions ¢(R) with an integral transformation "). Finally, basis functions 0"(R) are obtained with an angular-momentum projection of O(R) . These steps are, respectively, displayed in eqs. (17), (18) and (8) of ref '°). Basis functions ß ßw~,ï(R") involving distorted cluster wave functions can be obtained by combining (A"(R") with different oscillator parameters . The wave function (9) is approximated as (10)

D. 8aee ci aL / Nucleus-nucleus bremssirahlung

2S4

after selection of a finite set of N R"-values of the generatoe coordinate. The are determined from an eight-body Schradinger equation. As coefficients f 23 .22) eq. (10) cannot represent a scattering state, the microscopic R-matrix method is employed to connect it to a correct asymptotic behaviour. 2.3 . MATRIX ELEMENTS

The matrix elements of the hamiltonian calculated with the 0'"'(R") are obtained by projecting eq. (20) of ref. 24) on the orbital momentum L Here, we concentrate on the calculation of the reduced matrix elements( IP<< l I Jl EI ~',) ofthe E2 operator, appearing in eq. (4). For convergence reasons, the long-wavelength approximation 2 and Siegert theorem cannot be employed to simplify 'H2 [ref. )] . However, the microscopic R-matrix theory divides the configuration space in two parts . Convergence problems appear in the external part where antisymmetrization can be neglected. The matrix element of .112 can be separated into three components as in eq. (17) of ref. 2). This decomposition remains valid here, so that the problem reduces to the calculation of matrix elements, between basis functions ß'"(R), of the long-wavelength isoscalar operator, ,fi K=

e

s

'

2mck y i=i

( Pi - 8P~m),

(11)

where mN is the nucleon mass. This expression is invariant for galilean transformations. Since 0'"' (R) does not depend on the c.m . coordinate R...., the total momentum P... . can be replaced by its eigenvalue zero. In eq. (1i), the term containing R.... multiplied by Y; pi vanishes for the same reason. Hence, eq . (11) can be replaced by the simpler expression (12) Then the reduced matrix elements read l I (0'(R)I I~~l1E Jß"(R'))=32 r°(21+1) -'

E

( - )"(1'2 - W t~J10)

x~~dr o(OR')(4P(Rî)l lÙ w10(R'))sinOR , dOR. , 0

(13)

where R' lies in the xz plane at an angle BR, from the z-axis . The matrix elements appearing in eq. (13) involve the basis functions O(R) and operators such as '/2 eh _s $ a a _a ,f2 = . % 2zi -_xi -_yi ax MNek,, \ 16 Ir azi i i -, ( aY )

(14)

They are easily obtained as second-order polynomials in R, R' multiplied by spherical harmonics depending on OR , and gaussian factors in R, R' and cos B R . .

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The main difficulty consists in performing the Peierls-Yoccoz projection and the width transformations but these steps are performed algebraically on a computer. The integral over BR. in eq. (13) leads to expressions involving spherical Hankel functions. 3. Comparison of model calculations 3.1. DESCRIPTION OF THE MODELS

Our goal is to analyze the validity of bremsstrahlung as a probe of scattering wave functions. However, different authors have found that the bremsstrahlung 2,10,x). Hence, a meaningful comcross sections are very sensitive to phase shifts parison should rely on wave functions providing identical phase shifts. In addition, an experimental discrimination between the models requires that these phase shifts reproduce as accurately as possible the experimental ones. As we shall see, these conditions can be met in a rather satisfactory way. 25). With only A first model involves the deep gaussian potential of Buck et al. two parameters, it reproduces the experimental phase shifts in a very satisfactory way up to 15 MeV. Above 15 MeV, the experimental situation is less clear and several resonances are observed [see ref. 2°) for a summary]. Except in the vicinity of resonances, the deep-potential model (DPM) should remain valid up to at least 20 MeV. However, since the p+ 7Li channel opens at 17 MeV, the single-channel description becomes less accurate beyond that energy. A second potential model reproduces exactly the phase shifts ofthe DPM . Indeed, supersymmetry transformations 27'2x) allow one to construct potentials which are exactly phase equivalent to a given potential, but with different numbers of bound states. For a + a scattering, such a potential is presented in fig. 2 of ref. 27). It is rather similar to the shallow /-dependent potential of Ali and Bodmer 26) which has been employed in the context of bremsstrahlung by Langanke'"). However, the supersymmeri model (SPM) provides potentials which must be singular in order to satisfy the generalized Levinson theorem 29). In the following, we employ the SPM to construct a shallow potential, without unphysical bound states . To this end, two bound states for l = 0 and one for I = 2 are removed from the potential of Buck et al. For 1 :4, both potentials are identical. The microscopic model summarized in subsect. 2.2 has been applied in ref. 2°) to a description of a+a scattering involving monopolar distortion of the a-particles. A ten-channel approach was adopted corresponding to four a-states : the ground state, the giant monopole resonance and two pseudostates . We send the reader to the quoted reference for additional information. Here, we employ the same wave 32) functions. In ref. 2°), the parameter u of the Minnesota force is adjusted in order to reproduce the "Be ground-state energy. Then the s phase shifts closely agree with experiment . However, with that value, the 1= 2 and 4 resonance locations are slightly

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overestimated . In order to have phase shifts as close as possible in the different models, we adjust u separately for the different partial waves. The adopted values are u=0.9413 ((=0), u=0.9506 (1=2) and u=0.9600 (1_~ 4). With these values, the GCM phase shifts agree with the potential ones with an accuracy better than 3°. Further reducing this difference is not possible because of small differences in the curve shapes. As discussed below, this accuracy is sufficient for a meaningful comparison of models . A second microscopic model is also employed because of the length of the ten-channel calculation. A single-channel GCM calculation (with distorted aclusters) does not provide exactly the same phase shifts. However, if one readjusts the interaction parameter, a reasonable agreement can be obtained. In the singlechannel calculation, we employ u=0.977 (1=0), u=0.966 (1=2) and u=0.969 (1>>-4). The difference between the one- and ten-channel microscopic models will then essentially arise from differences in the wave functions at rather small distances. In order to have some estimate of the uncertainty on the cross sections due to phase-shift differences, we have performed several test calculations with this last model. When a common u=0.965 or 0.970 is taken for all partial waves, the phase-shift variation is smaller than 3° (except at resonances where it may reach 6°). Such values are of the order of or larger than the expected differences between the models. The bremsstrahlung cross sections obtained Lader these conditions differ by less than about 10% from those obtained with the fitted u-values . In some cases, the difference is even much smaller. Another hint on this uncertainty is given by the comparison between the single- and ten-channel models. Again, differences do not exceed 10% in general. Hence, from these simulations, we think that any difference larger than 10% in the cross sections will be physically significant. 3.2. CROSS SECTIONS IN THE HARVARD GEOMETRY

All experiments performed until now correspond to laboratory cross sections as defined in eq. (7) . In addition, the angles 0, and 0, are chosen equal . Since such experiments seem to be simpler to realize, we first consider them. In fig. 1, a comparison of the different models is presented under the conditions of the experiment realized by Peyer et aL ") (B,=0,=35°) . All the experimental points are obtained below 8;=10 MeV (c.m.) and most theoretical calculations of this cross section do not extend far beyond this energy z.K- '=) . In this energy range, the differences between the models are weak . However, the SPM provides a slightly larger cross section. This difference starts becoming apparent near the last experimental point and is of the order of magnitude of the error bar. In addition, the last experimental point might be too low. Therefore, discriminating between the models in that energy range is not possible . The situation is very different beyond 13 MeV. The SPM cross section is smaller than the DPM one by a factor of two. Even more interesting, the DPM and GCM provide different results . The difference must be

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Fig. 1 . Laboratory differential cross sections in the Harvard geometry at B,=02 =35°, as a function of the initial c.m. energy E; . The dashed line represents the single-channel GCM calculation and the crosses correspond to the ten-channel GCM calculation . Experimental data (circles) are from ref. ").

due to antisymmetrization effects. However, it is not large enough and would require improved statistics for the experimental points. Notice that the single- and tenchannel versions of the GCM provide essentially equivalent results. At 20 MeV, a monopole resonance should affect the results. This resonance is not described by the one-channel model. It is obtained in the ten-channel GCM [ref. 2°)], but near 25 MeV, and therefore does not appear in fig. 1 . The p+ 7Li channel which opens at 17 MeV is not described by the model . Can one increase the differences between the models? This can be done by exploiting the fact that the different phase shifts resonate . Since the I=0 resonance is very narrow, let us consider transitions between the l=4 resonance (E;= 11-12 MeV) and the I = 2 one (Er =3 MeV) "'). This leads to E,, =8.5 MeV. Assuming first B, = 02, eq. (5) provides 27°. This choice corresponds to Of= 90' [eq. (6)]. For 0 2 =27°, are there other possibilities? The situation is depicted in fig. 2 where lines corresponding to a constant Et and a constant E,, are represented as a function of 0, and E; . From this figure we deduce another interesting case: 82 =27° and B, =12°. The main difference is the value of of which is about 36°. In fig. 3, we present the Harvard cross sections for both cases. They are very similar and deserve the same comments . The cross sections are generally larger than in fig. 1 . Near 11 MeV, they display a marked peak which corresponds to both the initial 4+ and the final 2' resonances. At the peak energy, experiments should easily distinguish between the SPM and the other models . Note that the I=4 wave functions are identical for the DPM and SPM . Hence the large differences between the cross sections do not come from the resonant initial wave . The DPM differs by less than 15% from the GCM

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Zo o

L W

Fig. 2. Final energies Ef (full lines) and photon energies E, (dashed lines) as a function of the initial energy Ei and the angle B, for fixed BZ =27° in the Harvard geometry .

calculations which remain very close to each other. Discriminating between them again requires a higher energy. An advantage of both sets of angles in fig. 3 is that cross sections are larger . Again the accuracy of the model might be poorer beyond 17 MeV . 3.3. CENTRE-OF-MASS CROSS SECTIONS

Even with the help of figures such as fig. 2, laboratory cross sections are difficult to interpret. C.m. cross sections would be preferable . From fig. 3, we learn that transitions between resonances provide rather large cross sections with marked differences between the models . Are these differences due to the resonant character of the state or due to the large value of E,? What is the influence of the c.m. angle? In order to explore these questions, we now focus on a rather large initial energy. We choose E; =16 MeV, i.e. just below the first reaction threshold. At this energy, single-channel models remain a fair approximation z°). In fig. 4, transitions from E;=16 MeV to Er =3, 6 and 15 MeV are considered as a function of the c.m. angle Of. The final energies are respectively representative of a resonant final state, a large photon energy in a non-resonant case and a small photon energy. At small photon energies, the cross sections are large but the differences between the models are small. At large photon energies, the differences are much more important independently of the resonant or non-resonant character ofthe final state but cross sections towards a resonant state are significantly enhanced. Curiously, these conclusions do almost not depend on Of . However, depending on Er, some angles may lead to larger cross sections.

D. Baye et aL / Nucleus-nucleus hremsstrahlung

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N

N

b Iv

N L

C

Fig. 3. Laboratory differential cross sections in the Harvard geometry for tl, = Bz =27° and for B, =27°, Bz =12° as a function of the initial c.m. energy E; . The dashed lines represent the single-channel GCM calculation and the crosses the ten-channel calculation.

For two typical angles (30° and 90°), c.m. cross sections for E; = 16 MeV are displayed in fig. 5 as a function of Er . The 4' final resonance is visible as a broad bump in the 30° curves but not in the 90° curves. The 2' final resonance is apparent in both cases but is more important at 30°. Differences between models become significant for E;, ~,- 8 MeV (the SPM becomes different earlier). They are not directly related to the existence of a resonant behaviour but resonances are useful to increase the cross sections . The differences between the DPM and GCM are now very important and these models would clearly be discriminated by an experiment . The two versions of the GCM remain very similar at all energies . 4. Conclusions

The present work provides the achievement of a long effort : a multichannel microscopic model with monopolar distortion and an accurate treatment of the

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E i

16 MeV

=

------------------------------

10

90

60 ~4

(deg)

f

Fig. 4. Cross sections d-a/dE, d0, as a function of the c.m. deflection angle 0, at E;=16 MeV for the final energies E,=3, 6 and 15 MeV : single-channel GCM (dashed lines), DPM (full lines) and SPM (dotted lines).

Coulomb convergence problem. Although the differences with the results of our previous microscopic calculation -) may not appear spectacular, the present model represents an improvement in two respects . The separation of the Coulomb contribution allows us to restrict the calculation to a few partial waves for which the nuclear phase shift is not negligible . The use of distorted a wave functions solves the problem of inaccuracy of the phase shifts encountered in our earlier work. The improvement of the phase shifts, and of the wave functions, allows us to believe that we have reached an accurate description of a +a bremsstrahlung. Nucleus-nucleus bremsstrahlung should be a useful tool for probing scattering wave functions, with photon energies larger than about 10 MeV. This would be true v Y

18 14 i 12 10 . 8: 6h

-

900

V oÎ~ 4~ c. u

0

2

'2

i

0 2 /'le /'

4

6

8

°0

12

Fig. 5. Cross sections dzo/dE, df1, at two c.m. angles 8,=90° and 30` for E;=16 MeV as a function of the final energy E,: single-channel GCM (dashed lines), DPM (full lines) and SPM (dotted lines) . The crosses represent the ten-channel GCM results .

D. Baye et at / Nucleus-nucleus bremsstrahlung

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for different types oflaboratory or c.m. cross sections. Even the simplest experiments, in the Harvard geometry, would be useful for discriminating between models. Such experiments have already been realized 20 years ago but at slightly too low energies . Improving the experimental accuracy would be useful but not crucial in some cases. The different models we explore provide similar phase shifts . However, small phase-shift differences may introduce some uncertainty in the comparison . From simulations, we estimate the uncertainty on the cross sections due to inaccuracies of the phase shifts, to be smaller than 10%. Hence, only model differences larger than 10% are really significant (except for the comparison between the phaseequivalent DPM and SPM) . Since much larger differences are observed in many cases, experiments with an accuracy of that order would already be very useful . The shallow potential provides very different results from the other models . Since the SPM cross sections do not resemble the microscopic ones, i.e. those which are presumed to be the most realistic, we think that the SPM will be in clear contradiction with experiment. More interesting are the differences between the DPM and GCM. They are evidence for antisymmetrization effects in scattering wave functions which cannot be simulated by the unphysical bound states of the deep potential. On the contrary, differences due to the distortion introduced by closed channels are not very large . The ten-channel GCM represents the most elaborate model available for a+ a scattering wave functions. Improving this model would require big efforts. It is therefore very important to estimate its absolute accuracy. A significant disagreement with experiment below 17 MeV would be difficult to understand . The a+ a system is an excellent case for comparing wave functions. Only four partial waves (l=0-6) contribute significantly when the Coulomb term is treated separately ") and their effect may be enhanced by choosing energies where resonances occur. Even in that case, the behaviour of the cross sections cannot be explained in simple terms because the expression of the cross sections involves complicated interferences between partial waves at two different energies . Bremsstrahlung is more a powerful numerical way of testing wave functions than a simple physical model. Would the properties observed for a+ a remain valid when a larger number of partial waves contributes, i.e. for heavier systems? This question opens interesting research fields for theory . . . and for experiment. References 1) 2) 3) 4) 5) 6) 7) 8)

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