Deuteron-induced reactions in 12C at bombarding energies of 5 to 10 MeV

Nuclear Physics A127 (1969) 95--115; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

D E U T E R O N - I N D U C E D R E A C T I O N S I N 12C AT B O M B A R D I N G E N E R G I E S OF 5 TO 10 MeV H. C O R D S , G. U. D I N a n d B. A. R O B S O N

Research School of Physical Sciences, The Australian National University, Canberra, A.C.T. Received 19 December 1968

Abstract: Absolute differential cross sections for the reactions l~C(d, p)aaC leading to the ground state and the 3.09, 3.68 and 3.85 MeV excited states, 12C(d,g)l°B for the ground state and 0.717 MeV residual level and the elastic scattering of deuterons on xsC have been measured for bombarding energies of 5 to 10 MeV. An analysis of these measurements has been performed using the optical model and the distorted-wave and Hauser-Feshbach theories.

El

NUCLEAR REACTIONS lsC(d, d), (d, p), (d, ~), E = 5 to 10 MeV; measured o(E, 0), o(E; Ep, 0), o(E; E~, 0). Natural target.

1. Introduction Excitation functions for deuteron-induced reactions in light nuclei at low bombarding energies ( < 5 MeV) usually exhibit fluctuations typical of a compound-nucleus mechanism. In particular, 12C shows considerable structure even at much higher energies which probably arises because 12C is a more tightly bound nucleus than the average lp nucleus. On the other hand, the (d, p) angular distributions often have patterns which are associated with a direct-interaction process. These cross sections show a systematic trend as a function of energy with fluctuations about the average behaviour, and it is clear that in many cases there is a predominantly direct interaction mechanism interfering with a smaller compound nucleus process. While the distorted-wave theory 1) of (d, p) reactions has been applied very successfully to medium-weight and heavy nuclei at low incident energies, the theory cannot be expected to work as well for light nuclei. In these instances, the level densities in the compound nuclei are low, and the optical-model description may be rather poor because of insufficient averaging over resonance states. Moreover, the number of exit channels is usually small, therefore compound elastic scattering is not negligible. The ~2C nucleus is a notorious example of the above difficulties. However, Satchler 2) has shown that the over-all trend in the elastic scattering cross sections for the reaction 12C(d, d)12C between 3 and 34 MeV can be described in terms of an optical potential with parameters which vary smoothly with energy. This work ignored 95

96

H. CORDS et al.

the presence of compound elastic scattering. Hodgson and Wilmore 3) have studied the reactions of 1 to 5 MeV deuterons on ~2C; the compound elastic scattering cross sections were estimated using the Hauser-Feshbach statistical theory 4) with the transmission coefficients being obtained from appropriate optical potentials in the allowed channels. In this manner, they obtained better over-all agreement with the elastic scattering results and the (d, p) and (d, n) reaction data. This analysis gave satisfactory results at the higher energies, but poor agreement at the lowest energies where the compound nucleus contributions would appear to be dominant. This implies that the statistical assumption is invalid at these excitation energies of the 14N compound nucleus. Previous work 5-13) on deuteron-induced reactions in ~2C is not very extensive. In most cases, data were obtained at a few bombarding energies, and only relative reaction cross sections were measured. The purpose of the present experiment was to obtain more detailed information about the absolute cross sections in all the charged-particle open channels for incident energies between 5 and 10 MeV. No attempt was made to measure the (d, n) cross sections. The results have been analysed using the distorted-wave and Hauser-Feshbach theories as in the earlier work of Hodgson and Wilmore 3). However, the present analysis has two distinct advantages over the previous work. The bombarding energies are higher and energy-averaged cross sections have been used, therefore compound nucleus contributions to the various channels are smaller and arise from more levels in the compound nucleus. Similar theoretical analyses have recently been carried out for 160 [ref. ,4)] and 14N [ref. 15)] with considerable success.

2. Experimental procedure The measurements of the present work were performed with the Australian National University tandem accelerator. The deuteron beam was analysed in a 90 ° magnet and collimated in the beam line by two 2 mm diam. tantalum apertures spaced 31.5 cm apart. The energy calibration and the beam spread were known to approximately 0.1% and 5 keV, respectively. Self-supporting targets of natural carbon with thickness of 20 to 40 /~g/cm 2, which corresponded to an energy loss of 2 to 5 keV in the specified region of bombarding energies, were mounted in a 51 cm scattering chamber 16) at 45 ° to the beam direction. The charged-particle reaction products were detected by four silicon surface-barrier detectors placed at a distance of 18 cm from the target. The detection of different particles at several energies required the use of solid-state detectors of various depletion layer thicknesses. The angular resolution was less than 1°, while the error in the positioning of the counters was estimated to be less than +0.2 °. The electronics consisted of O R T E C 103 preamplifiers, ORTEC 203 amplifiers and multi-channel analysers. Pulse-height spectra were accumulated using two RIDL 400- and two RCL 512-channel analysers. The maximum deuteron beam current

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The extraction of excitation functions and angular distributions from the pulseheight spectra and the subsequent theoretical analysis were performed using an IBM 360/50 computer. Corrections due to dead-time of the analysers, background 1

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H. CORDS et al.

102

sian shaped line structure and a quadratic background term. Energetically coinciding deuteron and proton events in the spectra were separated by an additional measurement with an appropriate depletion layer thickness of the solid-state detector, which

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rejected the proton component s). A typical spectrum obtained at 9 MeV deuteron bombarding energy is shown in fig. 1. Absolute cross sections were determined by comparison with the known cross sections for the reactions 1 2 C ( p , po)~2C (g.s.) and 12C(p, pl)t2C (4.433 MeV state)•

tic(o, x)

103

The angular distributions were measured at 8.39 MeV proton bombarding energy. The resultant yield ratio reproduced the data of Barnard et al. ~7) within an estimated mean error of 0.7 %. A yield increase from the target of (2 4-1)% was found during a 16 h continuous bombardment• This carbon build-up effect was allowed for by assuming a linear dependence on the collected charge• Most of the measurements '

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were r e p e a t e d several times, a n d the a v e r a g e m e a n d e v i a t i o n f o r all particles, energies a n d angles was c a l c u l a t e d to be 3.9 %. M e a n d e v i a t i o n s o f u p to 10 ~ in s o m e cases were d u e to the p e a k - s e p a r a t i o n p r o c e d u r e . T h e e s t i m a t e d a v e r a g e m e a n e r r o r i n the a b s o l u t e cross sections is 16 9/0. 3. Experimental

results

Seven e x c i t a t i o n f u n c t i o n s were m e a s u r e d a t l a b angles 30 ° , 50 ° , 70 ° , 90 ° , 120 ° , 140 ° 1 6 0 ° f o r the r e a c t i o n s t 2 C ( d , d o ) t 2 C (g.s.), 12C(d, p o ) t 3 C (g.s.), t 2 C ( d , p t ) t 3 C

and

H. CORDS e t al.

104

(3.09 MeV state), 12C(d, pz)'3C (3.68 MeV state), 'zC(d, p3)tzc (3.85 MeV state), 12C(d, C
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7

8 9 I0 DEUTERON ENERGY ( M e V ) Fig. 6. Excitation functions for the lzC(d,p)lsc (6.87, 7.50, 7.55 and 7.68 MeV states), ltC(d,dl)1=C

(4.433 MeVstate) and ~=C(d,ct)~°B(2.15and 3.59 MeV state) reactions.The curvesare reduced (R = 0.3)compoundnucleuscontributionsfromthe Hauser-Feshbachcalculations.

Angular distributions were obtained for the same reactions at deuteron incident energies 5.00, 5.32, 5.41, 5.70, 6.05, 6.25, 6.40, 6.70, 6.90, 7.10, 7.18, 7.60, 7.85, 8.15 and 8.40 MeV. Most of the distributions were measured in I0 ° intervals except

"c(d, x)

105

for the two at 5.32 and 6.70 MeV, which were measured in steps of 5°. These data are presented in figs. 7-10 in the form of smooth solid curves. At the higher energies, several additional reaction channels became open. Limited excitation curves were measured for the reactions 12C(d, dl)12C (4.433 MeV state), 12C(d, p,)13C (6.87 MeV state), t2C(d, p5)13C (7.50 MeV state), t2C(d, p6)13C (7.55 MeV state), 12C(d, p~)t3C (7.68 MeV state), 12C(d, ct3)l°B (2.15 MeV state) and 12C(d, 0t,)l°B (3.59 MeV state) and are shown in fig. 6. Numerical values of the cross sections are available is).

'=C(d,d)'=C

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Fig. 7. Angular distributions for the arC(d, d)ltC elastic scattering reaction in mb/sr (centre of mass) represented by solid lines. 4. Theoretical analysis

The data for the reactions 12C(d, do)t2C (g.s.), 12C(d, po) 13C (g.s.), 12C(d, Pl)13 C (3.09 MeV state), 12C(d, p2)13C (3.68 MeV state), 12C(d, p3) ~3C (3.85 MeV state), 12C(d, *to)X°B (g.s.) and 12C(d, cq)t°B (0.717 MeV state) were averaged over two continuous energy intervals 5.00-6.70 MeV (Ed = 5.86 MeV) and 6.70-8.40 MeV (Ed = 7.49 MeV) in order to study the average behaviour of these reactions. The Hauser-Feshbach statistical theory 4) including the width fluctuation factor 19) was used to estimate the compound nucleus contribution to each reaction. These calculated cross sections were multiplied by a reduction factor R to allow for that portion of the incident flux which proceeds without compound nucleus formation. The shape elastic and direction reaction cross sections were computed using the optical-model and distorted-wave Born approximation 1) (DWBA) and were incoherently added

106

H. CORDS et al.

to the corresponding reduced compound nucleus contributions to give total cross sections. 12

ao

E

12 6 3 I

U

b 12

o

¢

b Jvv

Fig. 8. Angular distributions for the a2C(d, p0)tsC (g.s.) and t2C(d, pl)laC (3.09 MeV state) reactions in mb/sr (centre o f mass) represented by solid lines.

The deuteron nucleus interaction was assumed to be of the form

U (r) = C - Vg( V ) - i Wf(W) + Sr-l[dg (S)/dr]S. L + Mf(R)T~, where

g(i) = {l + e x p [(r-r,A~)/a,]} -', f ( i ) = 4[,q(i)] 2 exp [(r-r, A¢)/a,], TR = [ ( S ' r ) 2 r - 2 - - ~ ] , C is the Coulomb potential for a uniform charge distribution of radius R¢ = r¢A ~ and A the mass number of the nucleus.

ltC(d, x)

107

In a previous analysis 2o) of deuteron tensor polarizations measured at similar bombarding energies, a Tz. = [(S. L)Z + ½(S" L ) - ~L 2] tensor interaction was also included. However, a recent investigation 2~)of the elastic scattering and polarizations in +°Ca indicates that such a Tc term has a detrimental effect on the fits to

'2C(d,

32 16

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Fig. 9. Angular distributions for the x2C(d,p,)lsC 0.68 MeV state) and ltC(d,pD"C (3.85 MeV state) reactions in mb/sr (centre of mass) represented by solid lines.

the cross sections and vector polarizations. Consequently, the effect on the tensor polarizations of omitting Tc altogether was studied in the case of 12C. It was found that the strength 20) of this interaction (~ 1 MeV) was already so small that

108

H. CORDSet al.

TL could be neglected, TR being the dominant term required to describe the large tensor polarizations. Thus TL was ignored in the present analysis. Moreover, one should attempt to describe only the energy-averaged tensor polarizations. In the previous work, the data were perhaps fitted too closely on either side of the 6.4 MeV

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2C

b

Fig. 10. Angular distributions for the 12C(d, ~)X°B (g.s.) and ~tC(d, cq)l°B (0.717 MeV state) reactions in mb/sr (centre o f mass) represented by solid lines.

anomaly, thus leading to an over-estimate of the TR strength. It was found that the energy-averaged tensor polarizations (E a ~ 5.86 MeV) are satisfactorily described using potential D1 of table I. This potential was found by also fitting the elastic

"c(d, x)

109

TABLE 1 Optical-model potentials (a) Central potential Particle

Potential

V (MeV)

rv (fro)

av (fm)

W (MeV)

rw (fm)

aw (fro)

Ref.

deuteron

DI D2

130 125

0.90 0.90

0.90 0.90

4.16 6.79

2.10 2.10

0.50 0.50

3,20)

proton

P1 P2

51 49

1.25 1.25

0.65 0.65

7.00 7.00

1.25 1.25

0.47 0.47

22)

neutron

N1 N2

45 varied

1.32 1.25

0.66 0.65

9.00 0.00

1.26

0.47

22)

alpha particle

AI

80

2.07

0.55

4.00

2.07

0.30

2a)

r8 (fro)

a8 (fro)

M (MeV)

rR (fro)

aR (fm)

Ref.

--1.50

2.50

0.90

t,~)

(b) Spin dependent potential Particle

Potential

S (MeV" fm 2)

deuteron

D1 and D2

15

1.20

0.90

proton

PI and P2

16

1.25

0.65

22)

16

! .25

0.65

22)

E'-d" 5 . 8 6

MeV

-~d=7"49

R = 0.20

TI

R • 0'15

neutron

N2

8 4

2 1038

MeV

-

2

~ 1°28 d

.

4

•

",,

*

..-

.-'SE

_

T

2

cI lOt 8

...

,,

..

..

=_.

2 b

4

2

I0 o

0

50

I00

150 8

0

50

I00

150

C.M.

Fig. 11. Energy averaged angular distributions for the a2C(d, d)12C elastic scattering reaction at mean deuteron b o m b a r d i n g energies o f 5.86 MeV and 7.49 MeV compared with the c o m p o u n d elastic contributions (CE) from the Hauser-Feshbach calculations reduced by a factor R, the shape elastic contributions (SE) from the optical-model calculations and the totals (T = CE-i-SE).

1 10

H. CORDSeta[.

scattering data for E a = 5.86 MeV and is based on a potential previously obtained by Satchler 2). The Hauser-Feshbach cross sections were calculated using opticalmodel potentials of the form U(r) to obtain the transmission coefficients. For simplicity, only the central parts of the potentials DI, PI, N1 and A1 of table 1 were used in these calculations, and for a given final particle the same potential was employed for each channel irrespective of the excitation energy of the residual nucleus. In order to describe the elastic data at E d = 7.49 MeV (and the corresponding (d, p) I

I

I

I

1028

_

I

l

S=0"61 R'0"50

2

Po

I

I

S=0'19 R= 0"30

P2

_

4

2 1018 "~ m

•

*

T-

r - -

4 •

2 i0c8

:

............ : - "

\.o.."-" - . . . . . . "

."~,

-

°....._oo.°

.--="

I

4

,

I

,

I

'

I

I _

2

S=0'81 R= 0"50_

P3

1028 4

C=

"'",

2

T

"

1018 4 2

............................ _

".,..~

10° 8

,

0

I 50

,

I I00

,

@

- ~/~

I 150

i;-ii -

_

,--" "---I 0

,

I

50

,

I

I00

,

I

150

C.M.

Fig. 12. Energy averaged angular distributions for the lzC(d, p)laC (g.s., 3.09 MeV, 3.68 MeV and 3.85 MeV states) reactions at a mean deuteron bombarding energy of 5.86 MeV compared with the c o m p o u n d nucleus contributions (CN) from the Hauser-Feshbach calculations reduced by a factor R, the direct reaction contributions (DI) from the D W B A calculations and the totals (T = C N + D I ) .

measurements) the deuteron and proton parameters were varied slightly (potentials D2 and P2). Fig. 11 shows the elastic scattering results for reduction factors of 0.20 and 0.15, respectively. The only significant effect of including the T~ tensor is to fill in the minima near 100 °. Figs. 12 and 13 show the results for the (d, p) cross sections leading to the ½ground state and the ½+ 3.09 MeV, ~- 3.68 MeV and ~+ 3.85 MeV excited states, respectively. In each case, the DWBA cross section is the usual zero-range calculation

'~C(d, x)

i 11

for the potentials of table 1 multiplied by a finite-range correction factor of 1.65 and the appropriate theoretical spectroscopic factor S [refs. 24,2s)]. It was found that the tensor interaction TR had a negligible effect on the (d, p) cross sections, and this term was in general neglected. The neutron bound-state wave functions were obtained using potential N2 with the well depth being varied to give the correct asymptotic form. The calculated Hauser-Feshbach cross sections were multiplied by reduction factors, which gave the best over-all fits to the data when the direct and compound I

4 2 1018 4

'

I

I

..

-

"----,'

~

I

I

I

- -

S,, 0"19

R-o~o

-

R • 0"30

Po

,, "

-- - . ~ . ~ .

I

I

S,0.61

T~ ....

2

"'-:L............

I0o 8 4

-k,, F

• 1028

d

• •

/*

I

2 -\

"

..-.-=

•

......

"

-

,

I

'

I

"

I

,

I

,

.J

t

oO"

t

S ,0"97 - R , 0 " 4 0 - _ --

~

.........

6i

I

,

S • 0"81

-

R= 0.15

--_

4

2 cl I018 b

4

-

"..

T~

.

2 _- ..........

....:'::.-..-.;.~,

10° 8 .4

I 0

-~L

I

50

,

l

,

I00

I

,

150 #

I

0

,

I

50

,

I

I00

,

I

,

150

C.M.

Fig. 13. Energy averaged angular distributions for the l~C(d, p)lsc (g.s., 3.09 MeV, 3.68 MeV and 3.85 MeV states) reactions at a mean deuteron bombarding energy of 7.49 MeV compared with the compound nucleus contributions (CN) from the Hauscr-Fcshbach calculations reduced by a factor R, the direct-reaction contributions (DI) from the DWBA calculations and the totals (T = CN+DI). contributions were added incoherently. Reduction factors varying from 0.15 to 0.50 were obtained in this manner. It is seen that over-all theory gives a good description of the average cross sections. The most notable discrepancy is the underestimation of the (d, Po) cross section at forward angles for E d = 7.49 MeV. A similar difficulty has been encountered by others 12) for this reaction and probably arises from some non-statistical resonance contribution. The excitation functions for the do, Po, Pl, P2 and P3 reactions at lab angles of 30 °, 50 °, 70 °, 90 °, 120 °, 140 ° and 160 ° were calculated using optical-model parameters obtained by linear interpolation and extra-

112

rl. CORDS et al.

polation of those employed at 5.86 MeV and 7.49 MeV. Figs. 2-4 show that the average behaviour of these cross sections is in general satisfactorily described by the theoretical curves. The predicted elastic deuteron vector polarization was found to follow the average trend of the limited data available 26). These measurements suggest that the data should be averaged over a somewhat larger interval ( ~ few MeV) than the one adopted in the present work. The proton polarizations have been measured 27) for the (d, Po) and (d, Pl) reactions at 45 ° lab. It was found that although satisfactory agreement was obtained at E d = 7.49 MeV for the (d, pt) reaction, namely P = - 0 . 1 1 , the corresponding value 10 2

8

R=O.50

4

R-0.30 aI

a 0

2

"~ IO° 8

oooooooo

eo o

•

•

•

o°e°oo

o

-

4

oe

•

eo e

•

x. x.

//

CN

,J tO° 8 c~

4

b

2

"0

iO-I

0

50

100

150 0 50 I00 150 a C.M. Fig. 14. Energy averaged angular distributions for the lzC(d, ct0)l°B (g.s.) and x2C(d, ctt)~0B (0.717 MeV state) reactions at a mean deuteron b o m b a r d i n g energy o f 7.49 MeV compared with the comp o u n d nucleus contributions (CN) from the Hauser-Feshbach calculations reduced by a factor R.

of 0.66 for the ground state transition is in gross disagreement with the observed large negative values of m - 0 . 5 . However, at 45 ° lab the (d, Po) cross section has a large non-direct component which presumably contributes to the polarization because the statistical model has only limited validity. Originally it had been hoped to observe the j-dependence of the two 1 = 1 transitions, j = ½ for the ground state and j -- } for the 3.68 MeV level. However, the substantial compound nucleus contributions in these cases mask the effect to a large extent and make such a task very difficult. It was felt that further investigation of the j-dependence was not warranted because of the approximate nature of the compound nucleus calculations. Fig. 14 shows the (d, Cto) and (d, ctl) cross sections for F.a = 7.49 MeV and the corresponding Hauser-Feshbach results using reduction factors of 0.3. The alphaparticle potential AI is not very reliable, and while different parameters are unlikely to seriously affect the other channels, the predicted (d, :t) cross sections could be significantly modified. There is probably a substantial direct contribution to these

12C(d,x)

1i 3

cross sections, but these were not estimated because of the lack of any reliable theory for such calculations. Fig. 6 shows the excitation functions predicted for the limited data on the P4, Ps, P6 and P7 transitions assuming no direct components and reduction factors of 0.3. All these transitions lead to particle-unstable states, and the available distorted-wave computer programs did not allow calculations of the direct reaction cross sections in such cases. Moreover, the direct reaction is unlikely for both the p~ and P6 transitions, if the residual states have, as seems likely 2s), spins and parities of ~+ and ~-, respectively. The P5 and P6 cross sections were difficult to separate; therefore, it is interesting to note that the sum of the reduced Hauser-Feshbach cross sections is in good agreement with the sum of the measurements for these two reactions. Reduction factors of about 0.3 have been used in similar analyses for taN [ref. 15)] and 160 [ref. 1,)] and are also an average value of the reduction factors used earlier in the present work. For this value of R, the p, and P7 measurements are completely described by the statistical model. This is not the case for the remaining transitions d 1, O~3 and ~4. 5. Resonances

Although the general shapes of the angular distributions indicate the presence of direct processes, the excitation functions display a pronounced structure particularly at lower excitation energies and in the (d, ct) reactions. Two arguments will be pointed out in favour of some of the anomalies being associated with individual states in the compound nucleus. Firstly, some of the anomalies occur at different angles and in different reactions at the same excitation energy. Secondly, the ratio I-/D, where F is the mean level width and D the level density, is estimated to vary from 1.0 to 2.7 for deuteron bombarding energies of 5 to 10 MeV. These estimates were obtained by extrapolating from lower excitation energies and higher-mass nuclei and also using the HauserFeshbach transmission coefficients reduced by a factor of 0.3 and summed over all open channels according to the relation 2nF/D = 0.3 Z¢ To. Such values of F/D lead one to expect the possibility of some sharp resonances in the excitation functions and also suggest that the statistical model is not strictly valid at the energies under consideration. In general, the anomalies appear to be considerably broader in the (d, d) and (d, p) reaction channels than in the (d, ~t) channels, and large regions of strong excitation more than 1 MeV wide are observed. The most pronounced peaks in the (d, ~) excitation functions are found at bombarding energies of 5.35, 6.05, 6.40, 7.15 and 8.15 MeV (fig. 5). The F W H M is approximately 100 keV and 200 keV at the lowest and highest energies, respectively. The anomaly at 5.35 MeV is clearly seen in most of the reaction channels and has already been reported 29.30) as one or possibly two resonances. The 8.15 MeV anomaly can be correlated with peaks in the

114

H. CORDSet al.

three reaction channels Po, P3 and d o (figs. 2 and 3). N o particularly strong excitation with the expected width is observed at the energies 6.05, 6.40 and 7.15 MeV in the other reaction channels, although there is some evidence o f peaks at these energies in the P2 excitation function (fig. 4).

6. Conclusion The optical-model, distorted-wave and Hauser-Feshbach theories give a satisfactory over-all description of the present elastic scattering and the (d, p) data. The analysis shows that if shell-model spectroscopic factors are used for the direct reaction contributions, reduction factors o f a b o u t 0.3 are required for the calculated HauserFeshbach cross sections. For this value o f the reduction factor, the limited excitation functions for the P4, P5, P6 and P7 transitions are completely described by the statistical model. The requirement o f substantial c o m p o u n d nucleus contributions to the Po and P2 cross sections masks the expected j-dependence o f these I = 1 transitions. The estimated values o f F / D o f 1.0 to 2.7 indicate the possibility o f some sharp resonances in the excitation functions and suggest that the statistical model, while providing a g o o d estimate o f the c o m p o u n d nucleus contribution to each channel, is not strictly valid for the cases studied. The (d, ct) excitation functions exhibit p r o n o u n c e d peaks at b o m b a r d i n g energies o f 5.35, 6.05, 6.40, 7.15 and 8.15 MeV, but only the 5.35 and 8.15 MeV structures are strongly correlated in the other reaction channels. The authors wish to acknowledge the assistance o f Drs. M. lvanovich and B. V. N. R a o in the early stages of this experiment. They also wish to thank Dr. P. J. Dallimore for the use o f his Hauser-Feshbach c o m p u t e r program, Dr. F. C. Barker for several useful discussions and Professor E. W. Titterton for his interest and constant support during the experiment.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

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l 15

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