Experimental study of the ⁹Li breakup reaction on Pb target

Experimental study of the 9 Li breakup reaction on Pb target 1 2 3 corresp first-author weihuma@impcas.ac.cn weihuma@impcas.ac.cn weihuma@impcas.ac.cn 1 corresp jswang@impcas.ac.cn jswang@impcas.ac.cn jswang@impcas.ac.cn 1 1 1 1 1 1 1 1 1 1 2 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Lanzhou University, Lanzhou 730000, China Lanzhou University, Lanzhou 730000, China The breakup reaction of 9 Li on a Pb target has been measured at 32.7 MeV/nucleon for the first time. Two peaks of 6 He+t coincident fragments at 9.8 MeV and 12.5 MeV were observed and agree with the results of the GCM (generator coordinate method) calculation. The experiment was carried out thanks to a specially arranged detection system around zero degrees at the Heavy Ion Research Facility in Lanzhou, Radioactive Ion Beam Line in Lanzhou (HIRFL-RIBLL). Breakup reaction Nuclear cluster states Invariant mass method 1 Introduction 1 In recent years, with the development of RIB facilities, a large number of weakly bound nuclei are produced and have been extensively studied. Breakup reaction processes of weakly bound nuclei are important in the spectroscopy since the observables in the breakup reactions are strongly correlated to the microscopic structures of the ground and continuum states of weakly bound nuclei. These breakup reactions of weakly bound nuclei play important roles in researching the cluster resonance structure. Elucidating the mechanism of clustering in nuclei is of fundamental importance not only for nuclear many-body dynamics, but also for understanding some key processes in nuclear astrophysics [ 1 1 ]. The interest in nuclear clustering has been pushed strongly due to the study of neutron-rich and exotic weakly bound nuclei [ 2 2 ]. Many theoretical and experimental studies have been devoted to the cluster structure in nuclei in the past decades [ 3 3 - 9 9 3-9 ]. The missing mass method and invariant mass method can be applied to determine the excitation energy in the breakup reaction process. Several experimental studies of light nuclear clustering at HIRFL-RIBLL were performed. The RIBLL has been constructed at HIRFL and has been in operation since 1998. Consisting of four dipoles and sixteen quadrupoles, RIBLL is designed as a double-achromatic anti-symmetry separator with three focal points (F0, F2, and F4) and two focal planes (F1 and F3). RIBLL is a Projectile Fragmentation (PF) type facility with a large momentum acceptance and solid angle [ 10 10 , 11 11 ]. In a representative example carried out at HIRFL-RIBLL, new resonances close to the respective cluster separation thresholds were observed in 12 Be for the 4 He + 8 He and 6 He + 6 He decay channels based on an inelastic excitation experiment performed with a 12 Be beam at 29 MeV/u on a carbon target [ 12 12 , 13 13 ]. The strong clustering in 12 Be with 0 + state at 10.3 MeV is well demonstrated by using the model-independent angular correlation analysis and the extracted largely enhanced monopole transition matrix element of 7.0±1.0 fm 2 for this state. For the neutron-rich nucleus, 9 Li, many theoretical descriptions of the cluster structures have been studied, but experimental investigations are scarce. The cluster structures of 6 He+t, α +t+2n, and t+t+t in excited states of 9 Li were predicted by references [ 14 14 - 19 19 14-19 ]. In 1968, Ikeda et al. proposed a threshold rule for cluster formation, which suggests appearances of developed cluster states near the corresponding threshold energy [ 20 20 ]. They speculated that at the vicinity of an energy threshold for cluster separation, the nucleus tends to expand its size and favor the cluster formation in consequence [ 21 21 ]. From the point of view of Ikeda’s threshold rule, excitation energies of developed two-body cluster states in neutron-rich nuclei can be understood systematically. The possible appearance of 6 He+t cluster states near the 6 He+t threshold energy in excited states of 9 Li can be expected [ 17 17 ]. To describe various cluster and shell structures that may appear in the ground and excited states of light nuclei, studies of T. Suhara et al. [ 15 15 ] show that, in the prolate region along the γ = 0 o line, two-body cluster structures or linear-chainlike structures develop well as the deformation parameter, β , becomes large. They show a 6 He+t cluster configuration when ( β cos γ , β sin γ )= (1.03, 0.04). The present experiment was carried out thanks to a specially arranged detection system around zero degrees at HIRFL-RIBLL to observe the cluster structures in 9 Li through analyzing the coincident breakup fragments. 2 Description of the experiment 1 The primary beam is 12 C of 53.7 MeV/u delivered by the HIRFL. The RIBs were produced by bombarding the production target, Be with a thickness of 3038 μM, and analyzed and delivered to the secondary reaction chamber at the second focal point by RIBLL. We obtained a 32.7 MeV/nucleon secondary 9 Li beam with an intensity of about 1.1×10 3 particles per second and a purity of about 99%. The beam particle identification was realized event by event using the measured time-of-flight (TOF) and energy loss ( Δ E) values (see Fig. 2 Fig. 2 ). The Pb target was a self-supported foil with a thickness of 526.9 mg/cm 2 . In Fig. 1 Fig. 1 , a schematic view of the detector setup is given. Three position-sensitive parallel-plate avalanche counters (PPACs) [ 22 22 ] provided the position of the incoming beams with a position resolution better than 1 mm. Each PPAC had 51 gold-plated tungsten wires in both X and Y directions and a sensitive area of 50×50 mm. We successively placed the PPAC1, PPAC2, and PPAC3 at 1207 mm, 817 mm, and 112 mm before target. The position and incident angle of the beam particles at the target were determined by extrapolating the position information provided by PPACs event by event. A telescope consisted of two detectors and an E detector covering the angles 0° to 10°. The two detectors are double-sided silicon strip detectors (DSSDs) with a thickness of 523 μM for the DSSD1 and 527 μM for the DSSD2 and a sensitive area of 49×49 mm which is divided into 16 strips with the width of 3 mm and the interval of 0.1 mm between each two of them in each side. These two DSSDs were also applied to determine the position of the outgoing particles with a position resolution better than 3 mm. The E detector is a CsI(Tl) scintillator array which is composed of 8×8 CsI(Tl) crystals [ 23 23 ]. The active area of each scintillator unit is 21×21 mm 2 for the front side and 23×23 mm 2 for the back side, the length of each crystal is 5 cm. The DSSD1, DSSD2, and scintillator were successively placed at 40 mm, 133 mm, and 583 mm behind the target. A typical particle identification is given in Fig. 3 Fig. 3 where Δ E is the energy loss of particles in the detector DSSD2 and E is the residue energy deposited in the CsI(Tl) crystal. A clear separation of different nuclei is obtained. The shown nuclei p, d, t, α , 6 He, 7 Li, and 8 Li, are the breakup fragments from 9 Li on the Pb target. In our present investigation, we mainly take into account the two fragments recorded in coincidence. We applied the recorded positions by the detectors (PPACs, DSSDs, and CsI(Tl)) to reconstruct the tracks of the two charged fragments from the 9 Li breakup reaction on the Pb target. Energy calibrations of the telescope were achieved using beams of 9 Li, 6 He, 4 He, and 3 H produced from the 12 C primary beam. The energy resolutions are about 65 keV for DSSD1 and 47 keV for DSSD2 measured with a single-component ( 241 Am) and a three-component ( 241 Am- 238 Pu- 239 Pu) alpha source. The estimated energy resolution of the CsI scintillators are 7% ∼ 8% determined with a three-component ( 239 Pu- 241 Am- 244 Cm) alpha source. For a proton beam at the energy of 15 MeV, the resolution is 2.6%. A description of this kind of telescope array can be found in Refs. [ 24 24 , 25 25 ]. In the present work, the relative energy ( E rel ) of a pair of fragments was reconstructed from their kinetic energies ( T 1 , T 2 ) and the opening angle ( θ 12 ). According to the invariant mass method, the excitation energy of a resonance state is expressed as [ 26 26 ] <math display="block" id="M1"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>M</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1,2</mn></mrow></munder></mstyle><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>P</mi><mo>→</mo></mover><mi>i</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><msubsup><mi>M</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>M</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>M</mi><mn>1</mn></msub><mo>+</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>M</mi><mn>2</mn></msub><mo>+</mo><msub><mi>T</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><msqrt><mrow><mo stretchy="false">(</mo><msubsup><mi>T</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>M</mi><mn>1</mn></msub><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>T</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>M</mi><mn>2</mn></msub><msub><mi>T</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msqrt><mi>cos</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mn>12</mn></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>E</mi><mtext>x</mtext></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>M</mi><mo>−</mo><msub><mi>M</mi><mn>0</mn></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>E</mi><mrow><mtext>rel</mtext></mrow></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>E</mi><mtext>x</mtext></msub><mo>−</mo><msub><mi>E</mi><mrow><mtext>thres</mtext></mrow></msub><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where, M 0 is the rest mass of projectile; E thres is the threshold energy of the corresponding decay; E i and <math id="M2"><mrow><msub><mover accent="true"><mi>P</mi><mo>→</mo></mover><mi>i</mi></msub></mrow></math> are the energy and momentum of the fragment, i , respectively; E x is the excitation energy. The reconstructed excitation energy of 6 He+t decay is shown in Fig. 5 Fig. 5 . Through analyzing the telescope array, the coincident charged particles 6 He and t were identified and investigated. To estimate the resolution of the E rel and the detection efficiency (acceptance) for these 6 He+t events, the Monte Carlo simulation was performed considering the energy and spacial resolutions of the detectors. The simulation is plotted in Fig. 4(a) and (b) Fig. 4(a) and (b) . Where, the detection efficiency (peak at 37%) for the 6 He+t channel is shown in Fig. 4(a) Fig. 4(a) . The drop down of distribution of the detection efficiency in case of small relative energies is due to the ineffectiveness of the coincident measurement for 6 He+t fragments which enter into the same unit of the DSSDs or CsI(Tl) scintillator; the slow decrease of that (distribution of the detection efficiency) in case of large relative energies is due to one or both of coincident fragments may escape from the detector array with appreciable probabilities. As shown in the Fig. 4(b) Fig. 4(b) , the simulated resolution of the E rel is about 0.8 MeV at an E rel of 2.5 MeV and increases to 1.1 MeV at an E rel of 5.0 MeV. 3 Results and discussion 1 In the present experimental study of a 9 Li breakup reaction on Pb target, we reconstructed the excitation energy spectrum of 6 He+t decay. As displayed in Fig.5 Fig.5 , two resonance peaks just above the threshold energy (7.588 MeV) were obtained. We made a double-Gaussian function fitting and got the peaks at 9.8 MeV for the first and 12.5 MeV for the second. The GCM calculation for 6 He+t cluster states in 9 Li in Ref. [ 17 17 ] suggests that the energies of the third 3/2 - and the second 1/2 - states with relative orbital angular momentum L =1 lie within the range of the first peak at around 9.8 MeV, and those of the sencond 5/2 - and 7/2 - states with L =3 lie within the range of the second peak at around 12.5 MeV. The experimental values are slightly larger than the theoretical ones. However, it can be considered that the obtained values reproduce the data fairly well. Based on the present experimental analysis and the theoretical predictions, the picture of 6 He+t clustering just above the 6 He+t threshold energy in 9 Li might be supported. With considering the detection efficiency of the Δ E-E telescope and the effective incident beam particles at a target determined by extrapolating the position information provided by PPACs event by event, the experimental breakup cross section was deduced to be 1.5±0.3(stat.) mb for 6 He+t decaying channel. 4 Summary 1 In summary, the experiment of the 9 Li breakup reaction on a Pb target was carried out thanks to a specially arranged Δ E-E telescope system around zero degrees at the HIRFL-RIBLL. The breakup fragments p, d, t, 8 Li, 7 Li, 6 He, and α were identified by the Δ E-E telescope. Combining the analysis of tracks of the two charged fragments, coincident charged particles, 6 He and t, were identified and investigated. The excitation energy of 6 He+t decay was reconstructed by using the invariant mass method. Two resonant energies at 9.8 MeV and 12.5 MeV were observed, which agree with the GCM calculations. The present experimental analysis and the theoretical predictions might support the appearance of 6 He+t cluster states in 9 Li. References M. Chernykh , H. Feldmeier , T. Neff , et al . Structure of the Hoyle State in 12C . Phys. Rev. Lett ., 2007 , 98 : 032501 . DOI: 10.1103/PhysRevLett.98.032501 http://doi.org/10.1103/PhysRevLett.98.032501 Structure of the Hoyle State in 12C W. von Oertzen , M. Freer , Y. Kanada-En’yo . Nuclear clusters and nuclear molecules . Phys. Rep ., 2006 , 432 : 43 - 113 . DOI: 10.1016/j.physrep.2006.07.001 http://doi.org/10.1016/j.physrep.2006.07.001 Nuclear clusters and nuclear molecules M. Ito , N. Itagaki , H. Sakurai , et al . Coexistence of covalent superdeformation and molecular resonances in an unbound region of 12Be . Phys. Rev. Lett ., 2008 , 100 : 182502 . DOI: 10.1103/PhysRevLett.100.182502 http://doi.org/10.1103/PhysRevLett.100.182502 Coexistence of covalent superdeformation and molecular resonances in an unbound region of 12Be N. Soić , S. Blagus , M. Bogovac , et al . 6He + alpha clustering in 10Be . Europhys. Lett ., 1996 , 34 : 7 - 12 . DOI: 10.1209/epl/i1996-00407-y http://doi.org/10.1209/epl/i1996-00407-y 6He + alpha clustering in 10Be J. A. Liendo , N. Curtis , D. D. Caussyn , et al . Near threshold three-body final states in 7Li+7Li reactions at Elab = 34 MeV . Phys. Rev. C , 2002 , 65 : 034317 . DOI: 10.1103/PhysRevC.65.034317 http://doi.org/10.1103/PhysRevC.65.034317 Near threshold three-body final states in 7Li+7Li reactions at Elab = 34 MeV M. Milin , M. Zadro , S. Cherubini , et al . Sequential decay reactions induced by a 18 MeV 6He beam on 6Li and 7Li . Nucl. Phys. A , 2005 , 753 : 263 - 287 . DOI: 10.1016/j.nuclphysa.2005.02.154 http://doi.org/10.1016/j.nuclphysa.2005.02.154 Sequential decay reactions induced by a 18 MeV 6He beam on 6Li and 7Li M. Freer , E. Casarejos , L. Achouri , et al . α:2n:α Molecular band in 10Be . Phys. Rev. Lett ., 2006 , 96 : 042501 . DOI: 10.1103/PhysRevLett.96.042501 http://doi.org/10.1103/PhysRevLett.96.042501 α:2n:α Molecular band in 10Be M. Freer , J.C. Angélique , L. Axelsson , et al . Exotic Molecular States in 12Be . Phys. Rev. Lett ., 1999 , 82 : 1383 - 1386 . DOI: 10.1103/PhysRevLett.82.1383 http://doi.org/10.1103/PhysRevLett.82.1383 Exotic Molecular States in 12Be M. Freer , J. C. Angélique , L. Axelsson , et al . Helium breakup states in 10Be and 12Be . Phys. Rev. C , 2001 , 63 : 034301 . DOI: 10.1103/PhysRevC.63.034301 http://doi.org/10.1103/PhysRevC.63.034301 Helium breakup states in 10Be and 12Be Z. Sun , W. Zhan , Z. Guo , et al . Separation and identification of isotopes produced from 20Ne+Be reaction by radioactive ion beam line in Lanzhou . Chinese Physics Letters , 1998 , 15 : 790 . DOI: 10.1088/0256-307X/15/11/004 http://doi.org/10.1088/0256-307X/15/11/004 Separation and identification of isotopes produced from 20Ne+Be reaction by radioactive ion beam line in Lanzhou Z. Sun , W.-L. Zhan , Z.-Y. Guo , et al . RIBLL, the radioactive ion beam line in Lanzhou . Nucl. Instrum. Meth. A , 2003 , 503 : 496 . DOI: 10.1016/S0168-9002(03)01005-2 http://doi.org/10.1016/S0168-9002(03)01005-2 RIBLL, the radioactive ion beam line in Lanzhou Z.H. Yang , Y.L. Ye , Z.H. Li , et al . Observation of enhanced monopole strength and clustering in 12Be . Phys. Rev. Lett ., 2014 , 112 : 162501 . DOI: 10.1103/PhysRevLett.112.162501 http://doi.org/10.1103/PhysRevLett.112.162501 Observation of enhanced monopole strength and clustering in 12Be Z.H. Yang , Y.L. Ye , Z.H. Li , et al . Helium-helium clustering states in 12Be . Phys. Rev. C , 2015 , 91 : 024304 . DOI: 10.1103/PhysRevC.91.024304 http://doi.org/10.1103/PhysRevC.91.024304 Helium-helium clustering states in 12Be K. Arai , Y. Ogawa , Y. Suzuki , et al . Microscopic Multicluster Description of Light Exotic Nuclei with Stochastic Variational Method on Correlated Gaussians . Prog. Theor. Phys. Suppl ., 2001 , 142 : 97 - 156 . DOI: 10.1143/PTPS.142.97 http://doi.org/10.1143/PTPS.142.97 Microscopic Multicluster Description of Light Exotic Nuclei with Stochastic Variational Method on Correlated Gaussians T. Suhara , Y. Kanada-En’yo . Quadrupole Deformation β and γ Constraint in a framework of antisymmetrized molecular dynamics . Prog. Theor. Phys ., 2010 , 123 : 303 - 325 . DOI: 10.1143/PTP.123.303 http://doi.org/10.1143/PTP.123.303 Quadrupole Deformation β and γ Constraint in a framework of antisymmetrized molecular dynamics K. Muta , T. Furumoto , T. Ichikawa , et al . Three triton states in 9Li . Phys. Rev. C , 2011 , 84 : 034305 . DOI: 10.1103/PhysRevC.84.034305 http://doi.org/10.1103/PhysRevC.84.034305 Three triton states in 9Li Y. Kanada-En’yo , T. Suhara . 6He-triton cluster states in 9Li . Phys. Rev. C , 2012 , 85 : 024303 . DOI: 10.1103/PhysRevC.85.024303 http://doi.org/10.1103/PhysRevC.85.024303 6He-triton cluster states in 9Li T. Furumoto , T. Suhara , N. Itagaki . Investigation of the 3/2 <math id="M3"><mrow><msubsup><mrow/><mn>2</mn><mo>−</mo></msubsup></mrow></math> state of 9Li with microscopic structure and reaction models . Phys. Rev. C , 2013 , 87 : 064320 . DOI: 10.1103/PhysRevC.87.064320 http://doi.org/10.1103/PhysRevC.87.064320 Investigation of the 3/22− state of 9Li with microscopic structure and reaction models Y. Kanada-En’yo . Monopole transitions to cluster states in 10Be and 9Li . Phys. Rev. C , 2016 , 94 : 024326 . DOI: 10.1103/PhysRevC.94.024326 http://doi.org/10.1103/PhysRevC.94.024326 Monopole transitions to cluster states in 10Be and 9Li K. Ikeda , N. Takigawa , H. Horiuchi . The Systematic Structure-Change into the Molecule-like Structures in the Self-Conjugate 4n Nuclei . Prog. Theor. Phys. Suppl ., 1968 , E68 : 464 - 475 . DOI: 10.1143/PTPS.E68.464 http://doi.org/10.1143/PTPS.E68.464 The Systematic Structure-Change into the Molecule-like Structures in the Self-Conjugate 4n Nuclei K. Ikeda , H. Horiuchi , S. Saito . Chapter I. Introduction to Comprehensive Nuclear Structure Study Based on Cluster Correlations and Molecular Viewpoint . Prog. Theor. Phys. Suppl ., 1980 , 68 : 1 - 28 . DOI: 10.1143/PTPS.68.1 http://doi.org/10.1143/PTPS.68.1 Chapter I. Introduction to Comprehensive Nuclear Structure Study Based on Cluster Correlations and Molecular Viewpoint P. Ma , C.G. Lu , J.S. Wang , et al . Capability of PPAC in investigation of resonant properties in 18Ne via 17F+p . At. Energy Sci. Technol ., 2011 , 45 ( 3 ): 356 - 359 . (in Chinese) Capability of PPAC in investigation of resonant properties in 18Ne via 17F+p X. W. Yao , Y.Y. Yang , J.S. Wang , et al . Development of 8×8 CsI(Tl) array detector . At. Energy Sci. Technol ., 2010 , 44 ( 3 ): 358 - 361 . (in Chinese) Development of 8×8 CsI(Tl) array detector S.L. Jin , J.S. Wang , Y.Y. Yang , et al . A Δ-E telescope array for radioactive beam experiments . Nucl Instrum Methods Phys Res, Sect B , 2013 , 317 : 728 - 730 . DOI: 10.1016/j.nimb.2013.07.044 http://doi.org/10.1016/j.nimb.2013.07.044 A Δ-E telescope array for radioactive beam experiments S. L. Jin , J.S. Wang , M. Wang , et al . Application of ΔE-E telescope in 9C fragmentation reaction . At. Energy Sci. Technol ., 2012 , 46 : 385 - 389 . (in Chinese) Application of ΔE-E telescope in 9C fragmentation reaction Z. X. Cao , Y.L. Ye , J. Xiao , et al . Recoil proton tagged knockout reaction for 8He . Phys. Lett. B , 2012 , 707 : 46 - 51 . DOI: 10.1016/j.physletb.2011.12.009 http://doi.org/10.1016/j.physletb.2011.12.009 Recoil proton tagged knockout reaction for 8He The authors are very grateful to Dr. Y. Kanada-En’yo and C. J. Lin for valuable discussions. 10.1007/s41365-017-0334-4 This work was supported by the National Natural Science Foundation of China (Nos. U1432247 and 11575256), and the National Basic Research Program of China (973 Program) (Nos. 2014CB845405 and 2013CB83440x). 2017-02-15 2017-04-01 2017-04-11 2017-11-20 2017-12-01 2022-08-18 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2017 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-017-0334-4 http://dx.doi.org/10.1007/s41365-017-0334-4 . 2017 Nuclear Science and Techniques 12 28