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(3) STUDY OF THE 9L i(d,p)l0Li REACTION. A D issertation. Subm itted to the G rad u ate School of the University of N otre Dame in P artial Fulfillment of the Requirem ents for the Degree of. Doctor of Philosophy. by. Peter Angelo S anti, B.S., M.S.. JJ. •. Dr. )r. Jam es J. K I olata, D irector. D epartm ent of Physics N otre Dam e, Indiana Decem ber 2000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(4) UMI Number 9991258. UMI’ UMI Microfomn9991258 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(5) STUDY OF THE 9L i(d,p)10Li REACTION. A bstract by P eter Angelo Santi T he stru c tu re of th e particle unbound nucleus. 10Li, was investigated in a kine m atically com plete experim ent using th e 9L i(d,p)10Li reaction in inverse kinem atics a t an incident 9Li energy of 20 M eV /A . T h e experim ent utilized th e S800 Spec tro g rap h a t th e N ational Superconducting Cyclotron Laboratory to m easure the outgoing 9Li from th e breakup of 10Li in coincidence with the recoiling protons from the (d.p) reaction which were m easured using a series of silicon detectors. Based on the m easured kinem atics of the recoiling protons from the 9Li(d.p) reaction, a lower lim it to th e m ass of 10Li was m easured a t A = 33.098 ± 0.08 MeV which is consistent w ith previous measurem ents. A com plete reconstruction of the breakup of 10Li was perform ed based on the m easured p ro p erties of the outgoing 9Li nucleus, the recoiling proton, an d the in cident 9Li beam . T his reconstruction m ade it possible to isolate th e stru c tu re of 10Li associated writh a ground s ta te 9Li core from structure associated w ith a 9Li core in its first excited state. T he observed ratio of 9Li* core events to th e total num ber of I0Li events th a t were detected in th e experiment was 0.098 ± 0.04 at forward center of m ass angles (2.7° to 9.5°), and 0.244 ± 0 .0 4 a t m ore backward center of m ass angles (11° to 26°). T his ability to identify 10Li events associated w ith a 9Li ground s ta te core allowed for a relatively background free m easurem ent of the low-lying stru ctu re of I0Li. T he b est fit to th e Q-value sp ectra for 10Li events. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(6) P eter Angelo Santi w ith a 9Li ground s ta te core yielded a sta te located a t Q = -2.58(11) MeV which corresponds to a neutron sep aratio n energy S n = -0.35(11) MeV. Due to th e poor Q-value resolution th a t w as observed in this experim ent, however, th e existence of an additional low-lying s ta te a t Q > -2.43 MeV (S n > —0.2 MeV) could not be ruled out. T he angular d istrib u tio n of th is stru ctu re was m easured and com pared w ith coupled reaction channel (C R C ) calculations for an s-wave and a p-wave sta te . T he com parison between th e d a ta an d theory was inconclusive, however, in determ ining th e n atu re of th e observed stru c tu re in 10Li.. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(7) D edicated to my parents. Joe and Marie Santi.. ii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(8) CO NTENTS. TA BLES. ................................................................................................................................ v. F I G U R E S ................................................................................................................................. vi. A C K N O W LED G EM EN TS. ................................................................................................. xv. C H A P T E R 1: IN T R O D U C T IO N .................................................................................... 1. C H A P T E R 2: T H E O R E T IC A L B A C K G R O U N D .................................................... 6 2.1 Binding E n e r g y .................................................................................................... 6 2.2 Shell M o d e l ............................................................................................................... 12 2.3 Scattering T h e o r y .....................................................................................................15 2.4 Optical M o d e l........................................................................................................... 21 2.5 Coupled Channels C a lc u la tio n ............................................................................ 24 C H A P T E R 3: EX PER IM EN TA L M E T H O D .................................................................28 3.1 O verview ......................................................................................................................28 3.1.1 T he (d.p) reaction ...................................................................................29 3.1.2 K inem atics of th e 9L i(d,p)10Li R e a c tio n ............................................. 30 3.2 Beam P r o d u c ti o n .................................................................................................... 35 3.3 S800 Analysis L i n e .................................................................................................37 3.4 Target C h a m b e r ....................................................................................................... 47 3.4.1 CD d e t e c t o r s ............................................................................................. 52 3.4.2 Silicon S trip A r r a y ...................................................................................53 3.4.3 C alibrating th e silicon d e te c to r s ...........................................................58 3.5 S800 spectrograph .................................................................................................61 C H A P T E R 4: A N A L Y S IS .....................................................................................................71 4.1 O v erv iew ..................................................................................................................... 71 4.2 Analysis of E xperim ental S y s te m a tic s .............................................................. 74 4.2.1 S800 spectrograph .................................................................................. 74 4.2.2 Incident 9Li E nergy M e a su re m e n t....................................................... 82 iii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(9) 4.3. 4.2.3 Analysis lin e ................................................................................................... 90 4.2.4 Testing the transfer m aps ....................................................................... 99 4.2.5 13C (d .p )14C R e a c tio n ................................................................................ 102 Analysis of 9Li(d.p)10Li D ata ...........................................................................121 4.3.1 Mass m e a s u re m e n t....................................................................................129 4.3.2 Q-value m e asu rem en t................................................................................ 131. C H A P T E R 5: IN TER PR ETA TIO N AND CONCLUSIONS .................................. 137 5.1 Full R e c o n stru c tio n ............................................................................................... 137 5.2 M onte C arlo S i m u l a t i o n ...................................................................................... 150 5.2.1 Overview .................................................................................................... 150 5.2.2 Event G e n e ra tio n .......................................................................................151 5.2.3 D etection of sim ulated e v e n t s ...............................................................159 5.2.4 M onte Carlo r e s u lts ................................................................................... 163 5.2.5 F ittin g p r o c e d u re .......................................................................................175 5.3 A ngular D is tr ib u tio n ........................................................................................... 184 5.4 C o n c lu s io n s ............................................................................................................ 187 A P P E N D IX A: SILICON D E T E C T O R IN F O R M A T IO N .........................................190 B IB L IO G R A PH Y. .................................................................................................................192. iv. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(10) TABLES. 1.1. Sum m ation of th e previous studies of the s tru c tu re of 10L i .................. 3.1. Optical m odel p aram eters used in the FR E SC O calculation shown in Fig. 3.12......................................................................................................................... 50. 4.1. Results of w eighted fits to 5e v s a 803 for various direct beam runs w ith 1% m om entum spread (Runs 8 and 62) an d 1/4% m om entum spread (R un 6 1 )...........................................................................................................86. 4.2. Altered first ord er tran sfer m ap elements for th e analysis line................... 96. 4.3. Estim ated background in th e coincidence sp e c tra for th e strip detec tors due to th e 241 Am source................................................................................ 128. 5.1. Size of lim iting a p e rtu re s w ithin S800 sp ectro g rap h .................................... 160. 5.2. F itting p aram eters for th e CD Q-value sp e c tra ............................................. 184. A .l. Energy resolution for individual sectors of CD1 and CD2 as deter mined by m easuring th e w idth of the 3.18 MeV n-line in 148Gd. . . . 190. A .2. Energy resolution for th e individual strip d etecto rs based on the m ea sured w idth of th e 3.18 MeV a-line in 148G d ...................................................191. A .3. Average p o sition resolution of the different strip s on th e strip detec tors based on th e m easured w idth of the slits o n th e calibration mask. All resolutions given are FW HM in m m ............................................................191. v. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.. 5.

(11) FIG U RES. 1.1 C h a rt of th e nuclides in the region of p roton numbers from Z = 1 (H ydrogen) to Z = 6 (Carbon) showing th e location of the various types o f n eu tro n halo nuclei................................................................................. 3. 1.2 Schem atic diagram of the Borrom ean nucleus 11Li........................................ 4. 2.1 N eutron density distribution for a n eu tro n bound by 0.2 MeV (dashed curve) an d 8.0 MeV (solid curve) based on th e square well potential m odel........................................................................................................................... 11. 2.2 Level stru c tu re in 10Li based on naive shell m odel............................................13 2.3 P lot of th e calculated difference betw een th e excitation energy of the l / 2 + an d 1 /2 “ states and the n eu tro n separation energy for N = 7 isotones........................................................................................................................ 15. 2.4 (a) Typical radial wave function for a bound s ta te showing a positive scatte rin g length a. (b) Radial wave function for an unbound s ta te showing a negative scattering length.................................................................... 21 2.5 P lot of two sam ple (a) volume and (b) surface Woods Saxon p o ten tials. T h e solid curves in (a) and (b) are th e real and im aginary poten tials used to describe the 9Li + deuteron system, respectively, while th e dashed curves are the potentials used to describe th e 10Li -fp ro to n system . T he param eters for b o th sets of potentials are given in Table 3.1...................................................................................................................23 3.1. Schem atic diagram of the 9L i(d,p)10Li* => 9Li + n reaction perform ed in th e (a) lab fram e and (b) center of m ass reference fram e for the ( d p ) reaction. T he solid lines indicate th e quantities m easured in this experim ent whereas th e dashed lines are those quantities which were reconstructed from the d a ta .......................................................................... 32. 3.2. P lo t of th e calculated (a) l0Li lab angle and (b) proton lab angle as a function of th e 10Li center of mass scatterin g angle in th e 9L i( d p ) 10Li reaction based on an incident 9Li energy of 177 MeV. T he calculations were perform ed using the relativistic kinem atic code,P K IN ......................... 33. 3.3. D iagram of th e A1200 fragm ent se p a ra to r.......................................................... 36. vi. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(12) 3.4. D iagram of th e S800 analysis line and sp ectrograph .....................................38. 3.5. Plot of pad signal vs pad number in S803 for a single direct beam event..............................................................................................................................43. 3.6 P lot of pad signal vs pad number in S803 for a pulser run (a) before and (b) after th e pads were gain m atched. In this particular run. the pulser was set to half scale.....................................................................................44 3.7. D iagram of th e interm ediate image cham ber showing S803. S804 and the two m asks for a calibration run. T he imaging of the masks is carried o u t w ith E l as the trigger in the focal plane. For the actual d a ta runs, th e m asks were lowered completely o u t of the beam .................45. 3.8 D iagram of th e m asks used for calibrating (a) S803 and (b) S804.. . . 46. 3.9 R econstructed m ask p a tte rn for (a) S803 m ask and (b) S804 mask w ithout correcting for th e non-linear response of S803 and S804 in the d rift direction. T he arrows indicate where th e non-linear response of the detecto rs is m ost evident in the m ask pattern s. T he corrected m ask p a tte rn s are shown in (c) and (d) for S803 m ask and S804 mask, re s p e c tiv e ly ............................................................................................................... 48 3.10 D iagram of th e S800 targ et chamber. ............................................................. 49. 3.11 P lot of th e energy of the proton as a function of its scattering angle in the lab for th e 9Li(d,p) reaction (solid curve) in com parison w ith the kinem atics of th e lighter recoiling nuclei from p o ten tial background reactions betw een the 9Li beam and th e C D 2 targ et. T he arrows indicate the ran g e of angles covered by the silicon detectors in the targ et cham ber. All of th e kinematics were calculated using the code PKIN assum ing a 9Li incident energy of 19.7 M eV /A ...................................51 3.12 C alculated an g u lar distributions for populating an s-state (dark curve) or a p -state (red curve) in 10Li for the 9Li(d,p) reaction. T he dashed lines indicate th e range of angles covered by each of th e silicon detectors. 52 3.13 Block diagram of th e electronics for a CD detector. T he large arrows indicate m ultiple signals being processed through separate channels in the sam e m o d u le.................................................................................................. 54 3.14 Schem atic d iag ram of the Silicon Strip A rray as seen by th e target. In th e diagram , th e incident beam is coming o u t of th e page. Note th a t S trip D etecto r # 5 was not used in the experim ent.............................. 56 3.15 Block diagram of th e electronics for a S trip detector. T he large arrows indicate m ultiple signals being processed through separate channels in the sam e m o d u le.................................................................................................. 57. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(13) 3.16 (a) Schem atic diagram of th e m asks used to calibrate th e strip posi tion. T h e first three strips were located a t th e edge of th e d etecto r closest to th e targ et, (b) M easured m ask p attern along strip # 1 1 on S trip D etector # 3 . using the m eth o d described in equation 3.21. (c) M ask p a tte rn for strip # 1 1 on S trip D etector # 1 using m eth o d described in equation 3.22........................................................................................ 60 3.17 3.18. D iagram of S800 focal plane...................................................................................64 P lo t of (a) S801 TAC and (b) S802 TAC as a function of tim e as m easured by th e num ber of buffers w ritte n to tape sta rtin g w ith R un # 1 2 in com parison to the pad response in (c) S803 and (d) S804 over th a t sam e tim e period. T he plots correspond to the behavior of th e C R D C ’s over a period of approxim ately 75 hours in tim e. T h e arrow s indicate w hen R un # 1 8 and Rim # 3 4 occurred relative to R u n # 1 2 .. 67. 3.19 P lo t of th e initially m easured y-position in (a) S801 and (b) S802 as a function of th e num ber of buffers from R un # 18. T he solid line represents th e 5 th order polynom ial weighted fit used to described th e d rift as a function of buffer num ber. T he effects of th e correction are show n in (c) and (d) for S801 an d S802, r e s p e c tiv e ly ...........................68 3.20 P lo t of th e Ion C ham ber signal as a function of the (a) signal in E l and (b) BLT TAC for the direct 9Li beam as measured in th e focal plane o f th e S800...................................................................................................... 70. 4.1. D iagram of th e coordinate system used to m easure the m otion of a beam p article w ith m om entum p th ro u g h th e beam line............................... 74. 4.2. P lots of th e m agnetic field as a function of th e distance traveled inside a given elem ent for the four elem ents of th e spectrograph. T h e red solid curve is th e interpolated field while th e black dashed curve is th e fit using a 4th-order Enge function. T he residuals for each fit are p lo tte d directly above each m agnetic field plot. T he field settin g s shown were used w ithin the sp ectrograph to m easure th e recoiling 9Li from th e breakup of 10Li........................................................................................... 78. 4.3. D istrib u tio n of the reconstructed energy a t th e target for th e direct beam ru n w ith 1/4% m om entum slits in place..................................................81. 4.4. D istrib u tio n of th e reconstructed y position a t the targ et for th e hole ta rg e t rim from the measured focal plane inform ation....................................82. 4.5. R ay tra c e diagram of the S800 A nalysis line from the BLT to S804 for ray w ith th ree different m om enta, + 1/2% Sp (black). - 1/2% Sp (red), and th e central m om entum (green) of th e beam line. T h e d iagram was generated using COSY and assum es th a t the ray o riginated from a single p o in t a t the center of th e b eam line......................................................84. viii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(14) 4.6. Plot of dE v s a 803 for th e direct 9Li beam for R un # 8 . T h e solid curve represents th e linear w eighted fit used to determ ine the slope of SE in term s of a 803.............................................................................................................85. 4.7. D istribution of (a) <5a8o3 and (b) 6a803 — SE for th e 1/4% m om entum ru n ................................................................................................................................... 87. 4.8. Ray trace diagram of th e S800 Analysis line from th e PPA C to S804 for a series of rays spaced 3 m m ap art ab o u t the center of th e PPAC. a t different m o m enta.................................................................................................. 88. 4.9. Plot of a 803 as a function of PPA C x for R un # 8 ..............................................89. 4.10 (a) D istribution of th e initially reconstructed x position a t th e target for the hole targ e t run using th e COSY transfer m ap for th e 'hole ta rg e t’ run. (b) D istrib u tio n of th e reconstructed y position a t th e target using the C O SY transfer m ap for the ‘hole ta r g e t’ rim . T he black histogram is th e forward reconstruction using inform ation from S803 and S804 while th e purple histogram is inverse reconstruction from Figure 4.4.............................................................................................................92 4.11 P lot of th e difference betw een reconstructed y positions a t th e target ( A ytarget) as a function of (a) i/803 and (b) 6803 ............................................... 94. 4.12 Plot of the difference betw een reconstructed y positions a t th e targ et (A ytarget) as a function of 6803 (top row) and y803 (b o tto m row) for the (a) initial tra n sfer m ap elem ents, (b) final (y\b) elem ent b u t initial (y \y ) element, an d (c) final (y\b) and (y\y) elem ents included in the transfer m ap. T he horizontal line indicates no co rrelation between th e two param eters..................................................................................................... 95 4.13 (a) R econstructed x t a r g e t d istrib u tio n using the corrected forw ard m ap for the ‘hole ta rg e t’ run. (b) R econstructed y t a r g e t d istrib u tio n for the ‘hole ta rg e t’ ru n ........................................................................................................... 97 4.14 (a) Reconstructed y t a r g e t d istrib u tio n using the corrected forw ard m ap (black histogram ) an d th e inverse transfer m ap (purple histogram ) for the ‘hole ta rg e t’ run. (b) A y target distribution for th e hole ta rg e t run. (c) A atarget d istrib u tio n and (d) A b target d istrib u tio n for th e ‘hole ta rg e t’ ru n ......................................................................................................................98 4.15 (a) Plot of bf o c a i vs a f o c a i as m easured in the focal plane for R un # 51. (b) P lot of x f o c a i vs y f o c a i as measured in the focal plane for Run # 51. (c) P lo t of th e reconstructed bf0cai vs dfocai. (d) P lo t of the reconstructed x f o c a i vs y f o c a l ......................................................................................................................................101 4.16 Difference d istrib u tions betw een reconstructed and m easured p aram eters in the focal p lan e............................................................................................ 102 4.17 Level diagram for 14C .............................................................................................. 104. ix. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(15) 4.18 (a) P lot of A E vs E from th e focal plane for the direct 13C beam , (b) P lot of A E vs E for th e reaction products from th e 13C (d ,p )I4C reaction, (c) Plot of A E vs E for reaction events in coincidence w ith the silicon detector array. T he cross in (b) and (c) indicates the location of th e direct 13C beam for reference...................................................106 4.19 (a) P lot of energy vs angle for th e I4C events in coincidence w ith th e silicon detectors, (b) P lo t of energy vs angle for all of th e p ro to n events in coincidence w ith th e focal plane. The solid curves are th e calculated kinem atics from P K IN for the ground state and 1st excited s ta te (E* = 6.093 MeV) of 14C .............................................................................108 4.20 P lot of 14C energy vs proton angle for the coincidence events. T he solid curves are the calculated kinem atics from PKIN for th e ground s ta te and 1st excited s ta te in l4C ........................................................................ 109 4.21 (a) P lo t of the 14C energy vs angle in the focal plane showing th e gate used to identify th e ground s ta te band in the silicon detectors, (b) P lo t of the proton energy as a function of angle for the g ated ground s ta te events in th e focal plane. T h e solid curve represents th e ideal kinem atics of th e p roton while the dashed and d o tted curve correspond to the expected kinem atics for the punch th rough events assum ing th e protons passed th ro u g h th e center of th e d etecto r or th e edge of th e detector, re s p e c tiv e ly .............................................................. I l l 4.22 (a) P ro to n kinem atic plots for th e various strip detectors correspond ing to th e excited states in 14C. T he solid curve in each plot is th e expected kinem atic curve for th e population of th e 1st excited s ta te in 14C while the dashed curve corresponds to the threshold for th e breakup of 14C. (b) C orresponding Q-value histogram s for th e various strip d etecto rs............................................................................................................113 4.23 (a) P ro to n kinem atic plots for th e various strip detectors correspond ing to th e excited states in 14C w ith array offset in the x direction by 1.1 cm. (b) C orresponding Q-value histogram s for the strip detectors w ith th e offset............................................................................................................114 4.24 P lo t of th e proton energy vs angle for th e ground sta te events in th e strip detectors with the 1.1 cm x offset in place.............................................115 4.25 (a) P lo t of th e m easured Q-value in CD2 as a function of the sector num ber. T he dashed line indicates the calculated Q-value for th e ground s ta te in 14C. (b) Q-value sp ectra from CD2...................................... 116 4.26 (a) P lo t of th e m easured Q-value in CD2 as a function of th e sector num ber w ith CD2 offset from th e optical axis, (b) C orrected Q -value sp ec tra as m easured by C D 2.................................................................................117. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(16) 4.27 (a) Com parison of th e m easured Q-value sp ectra in CD1 (solid his togram ) w ith th e m easured Q-value sp ectra in CD2 (dashed his togram ). (b) P lot of th e m easured proton energy’ vs angle for C D l (circle) and CD2 (square). T he solid curves represent constant Qvalue. (c) C orrected Q-value spectra for the energy-shifted C D l (solid histogram ) com pared w ith the m easured Q-value spectra in CD2 (dashed histogram ), (d) M easured p roton energy vs angle for energy shifted C D l (circle) and CD2 (square)................................................120 4.28 (a) P lo t of the A E vs E sp ectra of the reaction p roducts from the 9L i(d ,p )10Li reaction, (b) P lot of the A E vs E sp ectra for reaction events in coincidence w ith a silicon detector................................................... 122 4.29 (a) P lot of the silicon energy as a function of angle for all of the events detected by th e silicon detectors, (b) P lot of th e silicon energy' as a function of angle for events in coincidence w ith 9Li in the focal plane. The solid line indicates th e calculated kinem atic edge for the 9L i(d .p )l0Li reaction................................................................................................123 4.30 (a) P lo t of the p roton energy as a function of buffer num ber for (a) S trip # 2 (b) S trip # 3 and (c) S trip # 6 for th e 9L i(d ,p )10Li reaction runs. The arrow indicates the s ta rt of R un # 25.......................................... 125 4.31 Energy spectrum of th e 241Am source as m easured by (a) S trip # 2 , (b) S trip # 4 , and (c) S trip # 6 in singles mode. T he solid line indicates th e lower edge of th e alp h a peak used to sep arate th e alpha peak from th e flat background..................................................................................................127 4.32 Plot of calculated u n certainty in the Q-value as a function of the pro ton scattering angle w ith the m easurem ent of th e incident included (solid curve) and w ithout a m easured incident energy (dashed curve). B oth calculations were perform ed for a Q-value of -2.25 MeV assum ing an uncertainty in th e m easured proton energy of ± 1.5% (<r) for all of th e scattering angles.....................................................................................130 4.33 P lot of proton energy vs proton angle for d a ta in coincidence with 9Li in focal plane, S803, an d S804. The solid curve is th e best kinem atic fit to the d a ta varying only th e mass of 10Li while th e gold points indicate the events which were considered in fitting th e kinematic edge. In the inset, th e reduced y 2 for the kinem atic fit as a function of the mass excess of 10Li...................................................................................... 132 4.34 (a)-(b) Q-value sp e c tra for th e CD detectors an d S trip detectors, respectively, for all coincident proton events, (c)-(d) Q value spectra for CD detectors and S trip detectors, respectively, for proton d a ta in coincidence w ith S803 and S804 as well as th e focal plane. In all of the figures, the d o tte d line is Q-value corresponding to the 9L i+ n threshold while th e dashed line is the Q-value corresponding to the m easured mass of 10Li.............................................................................................133. xi. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(17) 4.35 R elative an g u lar distribution of th e (d.p) events w ith Q > -4.0 M eV as a function of (a) the p ro to n la b o rato ry scattering angle a n d (b) th e 10Li center of mass angle................................................................................. 136 5.1. P lo t o f (3 as a function of th e m ass of l0Li. T h e value was calculated assum ing a n incident 9Li energy of 177.09 MeV and a detected p ro to n energy of 2.4 MeV. The dashed curve represents a 2 cr error in th e determ in ed 10Li to ta l energy assum ing th a t th e energy of th e incident 9Li was d etected. The d o tted curve is th e error w ithout an incident energy m easurem ent................................................................................................. 140. 5.2. Sum m ed reconstructed breakup Q -value sp ectra for th e (a) C D de tectors an d th e (b) Strip detectors. In b o th (a) and (b). th e dash ed histo g ram is th e d istribution of events in which the Q-value for th e (d.p) reactio n was m easured larger th a n -2.20 MeV......................................142. 5.3. R econstructed Q-value spectra for th e breakup of 10Li as a function of th e Q -value for th e (d.p) reaction as m easured by (a) th e CD d etecto rs and th e (b) strip detectors. In b o th figures, the dashed line indicates th e reactio n Q -value corresponding to th e measured 10Li m ass while th e d o tte d line indicates th e reaction Q-value corresponding to th e 9Li -f- n th re sh o ld ...................................................................................................... 144. 5.4. (a) Qbreak vs Q-value for th e S trip d etecto r d a ta from Fig. 5.3(b). (b) P ro to n E nergy vs Proton Angle for th e S trip data. In both cases, th e solid d o ts indicate the events which do n ot have the proper co rrelation betw een Qbreak and the reaction Q -value...........................................................145. 5.5. atarget vs btarget as reconstructed from th e spectrograph inform ation for th e outgoing 9Li m easured in coincidence w ith the proton d etected in (a) S trip # 1 . (b) Strip # 2 . an d (c) S trip # 4 . In each figure, th e black filled circles are events w ith th e (d.p) Q-value > -4.1 MeV, th e gold filled circles are events w ith Q-values ranging from -4.1 M eV down to -6.3 MeV. and th e open black circles are events w ith Q < -6.3 M eV .......................................................................................................................148. 5.6. Q-value o f th e (d.p) reaction as a function of 10Li laboratory breakup angle 9br for (a) th e CD and (b) th e S trip detectors. T he solid d o ts in (b) are th e uncorrelated events w hich were selected in Fig. 5.4 . . . 149. 5.7. (a) Illu stratio n of th e rejection m eth o d in producing th e ainc d istrib u tion. (b) R esu ltan t ainr distrib u tio n from th e M onte Carlo calculation (solid line) com pared with the m easured a*nc distribution from a d i rect b eam ru n (red circles)..................................................................................... 153. xii. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(18) 5.8. (a) M easured binc as a function of yinc for th e direct beam Run # 6 2 . (b) R otated binc as a function of th e ro ta te d i/,„c used to determ ine the projections of th e two variables onto th e ro ta te d axes, (c) bmc vs yinc for 2500 events generated in th e M onte Carlo sim ulation. T he line in Figures (a) and (c) represents th e slope used to determ ine the angle for ro ta tin g th e spectrum ........................................................................... 155. 5.9. Com parison betw een d a ta (black curve) and th e M onte Carlo results (gold curve) for (a) x 80i (b) ymi (c) a /oca/ an d (d) bfocaL for a run w ith no targ et in place and th e spectrograph set to m easure the direct 9Li begun. In all of th e figures, the d a ta have been scaled down by a factor of 15 to m atch th e M onte Carlo results................................................. 161. 5.10 Com parison betw een d a ta (blue histogram ) an d M onte Carlo results (gold histogram ) for (a) x 80i (b) ysoi (c) a focai and (d) bjocai with the CD2 targ et in place. In all of th e figures, th e sim ulated results have been scaled to m atch th e measured d a ta ..........................................................164 5.11 Sim ulated excitation energy of I0Li E x as a function of the kinetic en ergy of th e outgoing 9Li for events in which th e corresponding proton was detected by (a) th e CD detectors and (b) S trip detectors. T he solid gold circles indicate events which were detected by S801 and S802, th e open black circles indicate events which were not detected in the focal plane, and th e open green squares are th e events which hit th e beam block in front of S801....................................................................167 5.12 (a)-(b)R eaction Q-value as a function of 9br for th e CD and S trip detectors, respectively. In the figures, the red circles are the d a ta gated based on th e results of the M onte C arlo sim ulation which as sumed the 9Li core in its ground s ta te (light blue circles), while the blue open circles indicate th e d a ta th a t were n o t included in the gate. (c)-(d) Qbreak as a function of the reaction Q-value for the CD and Strip detectors, re s p e c tiv e ly ............................................................................... 169 5.13 (a)-(b) G ated reaction Q-value distributions for the CD and Strip de tectors, respectively, (black circles) along w ith th e normalized results of the M onte Carlo sim ulation, (c)-(d) G ated Qbreak distributions for the CD and S trip detectors, respectively, w ith th e corresponding normalized M onte C arlo results...........................................................................171 5.14 (a)-(b)R eaction Q-value as a function of 9br for th e CD and S trip detectors, respectively. T he brown circles are th e result of the sim ulation where a 9Li* was produced in the breakup, while the yellow circles indicate th e sim ulated results producing 9Li in its ground s ta te shown in Fig. 5.12. (c)-(d) Qbreak as a function of the reaction Q-value for the CD an d S trip detectors, re s p e c tiv e ly ................................................ 173. xm. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission..

(19) 5.15 (a)-(b) 9Li* g ated reaction Q-value distributions for the CD an d S trip detectors, respectively, (black circles) along w ith the norm alized re sults of th e M onte Carlo sim ulation, (c)-(d) 9Li* gated Qbreak d istri butions for th e CD and Strip detectors, respectively, with th e corre sponding norm alized Monte Carlo results........................................................ 175 5.16 (a)-(b) Sim ulated Q-value spectra for the CD and Strip detectors, respectively, (c)-(d) Simulated Qbreak spectra for the CD and S trip detectors, respectively. In b o th cases, a delta function was assum ed for th e excitatio n of I0Li centered a t Ex = 0.5 MeV. The solid lines in th e figures are th e results of fits to the sim ulation using G aussian d istrib u tio n s...............................................................................................................178 5.17 G round s ta te Q-value spectrum of the 13C (d ,p )l4C as m easured by the CD2 detector compared w ith the results from the M onte C arlo sim ulation normalized by the num ber of events in each spectrum . . . 179 5.18 (a)-(b) Sim ulated Qbreak spectra for the CD and Strip detectors, re spectively, assum ing a delta function excitation curve located a t E x = 0.2 MeV (solid histogram) and E x = 1.2 MeV (dashed histogram ). (c)-(d) Sim ulated Q-value spectra for th e CD and Strip detectors, respectively, for E x = 0 .2 MeV (solid histogram ) and E x = 1.2 MeV (dashed h isto g ram )..................................................................................................182 5.19 F its to th e CD Q-value spectrum assum ing a single resonance (solid curve) an d 2 resonances (dashed curve) along w ith an uncorrelated background (d o tted curve). T he p aram eter for the fits are presented in Table 5.2................................................................................................................ 183 5.20 Efficiency corrected relative angular distribution of the (d,p) events w ith Q > -4.0 MeV as a function of (a) the proton laboratory scat tering angle and (b) the 10Li center of mass angle. The black and red curves in (b) are the FRESCO calculations for the s-wave a n d pwave, respectively. The calculations have been normalized to m atch the d a ta based on the first d a ta p o in t............................................................... 186. xiv. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(20) A CKNOW LEDGEM ENTS. First and forem ost. I m ust th an k my advisor. Dr.. Jam es J. K olata. whose. wisdom, su p p o rt, patience, and confidence in me m ade this thesis possible. As w ith all projects in Nuclear physics, my thesis experim ent was the result of a collaborative effort.. I m ust thank the people a t M ichigan S tate University,. specifically B rad Sherrill, whose work on the A1200 fragm ent separator and the S800 S pectrograph was crucial for this experim ent, D aniel B azin who designed th e detector array used in this experim ent and whose knowledge of the S800 Spectro graph was quite helpful in understanding the d ata, and Jo h n Yurkon. who designed and m aintained th e d etectors used in the S800 S pectrograph as well as the tracking detectors. I would be remiss if I d id n ’t th an k my fellow group-m ates here at N otre Dame, who I feel are m ore like friends th a n colleagues. Specifically, Don Peterson, w ith whom I have had m any conversations about the difficulties w ith our d ata. Valdir G uim araes, and Ryan W hite-Stevens, who assisted in th e initial analysis of this data. I would like to th a n k P aul DeYoung, G raham Peaslee, and Pete Jolivette from Hope College who provided their tim e and talents during th e ru nning of the exper im ent and helped keep m e on track. I appreciate the tim e we have spent together working on experim ents an d hope to do so again in th e future. I would also like to acknowledge the staff a t N otre D am e’s Nuclear S tru ctu re Laboratory: Jim Kaiser, whose su p p o rt of the lab ’s com puter cluster was crucial for xv. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(21) th is work as well as m any others a t N otre D am e: L arry Lamm, w ith w hom I have h ad many productive an d entertaining conversations, and Charles Guy. O f course life is n o t w orth living w ithout people to love, and so I m ust th an k my wife. Naomi, whose unconditional love an d su p p o rt helped get th ro u g h all of th e rocky tim es in m y life in the last few years. I also must th an k my son. Joseph M ordecai, for giving m e the burst of energy th a t was needed to com plete th is thesis an d focus on the task s a t hand. Finally. I m ust th a n k my parents. Joe an d M arie Santi. who gave m y th e perfect exam ple of how to live in this world, and who ta u g h t me everything chat is im p o rta n t in life. I am afraid m y g ratitude to them can never be expressed in this lifetime. I dedicate this thesis to them and to their m em ory. M ay they rest in peace.. xvi. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(22) C H A PT E R 1 IN TR O D U CTIO N. Since its discovery by R utherford in 1911 [1]. the stu d y of the atom ic nucleus has advanced considerably over th e past century. A vast num ber of stable and radioac tive nuclides have been studied and cataloged, allowing for a general understanding of th e basic properties of th e nucleus as well as how these properties vary as a func tion of th e n eutron and p roton num ber. W hile much is known ab o u t the nucleus, m any questions still rem ain in term s of the behavior of th e nucleus as it is pushed to th e extrem e lim its of stability.. It is not clear, for example, if the properties. th a t are observed in stable nuclei rem ain the sam e for those nuclei which are barely bound against particle decay. It is also unclear how m uch the current models of th e nucleus th a t were developed to describe the properties of stable nuclei will need to be adjusted for nuclei which are weakly bound. It is these questions, am ong m any others, which need to be addressed in order to enhance our understanding of th e atom ic nucleus. Until recently, th e stu d y of th e nucleus was lim ited to those nuclei near th e val ley of stability due in p a rt to th e fact th a t the m ajo rity of the beam s and ta rg ets available for use in experim ents consisted of nuclei which were either stable or had extrem ely long lifetimes. T his restriction in the incident beam lim ited the ability of earlier experim ents to produce nuclei which were far from stability. W ith th e developm ent of radioactive ion beam s (RIB), nuclei which exist far from stab ility. 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(23) could now be produced by various experim ental techniques and studied sy stem at ically. T he subsequent expansion in th e range of nuclei which are accessible as a result of th e developm ent and usage of R IB s has led to the discovery' of new and exotic properties of th e nucleus. O ne of th e more interesting discoveries th a t has occurred th ro u g h th e use of RIBs is the existence of a "neutron halo" in weakly bound nuclei n ear th e neutron drip line, w here neutron rich nuclei reach th e lim it of particle stability. This neutron halo consists of one or more n eutrons th a t exist a t rather large distances away from the center of th e nucleus, which is evident as an extended tail in th e n eutron density d istrib u tio n [2]. As a result of th is tail in the neutron density d istribution, halo nuclei have a more diffuse surface region as well as a larger radius th a n is observed for m ost nuclei. T he increased size of th e halo nucleus relative to o th er nuclei generally increases the probability of th e halo nucleus interacting w ith o th er nuclei as is evident by large interaction cross sections [3]. Figure 1.1 shows th e location of the various neutron halo nuclei th a t have been discovered relative to th e valley of stab ility in the chart of the nuclides. T he m ost common type of halo nucleus th a t has been discovered to d ate is th e tw o n eu tro n halo nucleus such as 11Li, 6He and 14Be. T h e two neutron halo nuclei th a t have been discovered to date, with the exception of 12Be. exhibit th e unique p ro p erty of being stable against particle decay while th e nucleus w ith one less n eu tro n is particle unstable. This implies th a t removing one n eu tro n from a two n eu tro n halo nucleus produces a particle u nstable nucleus, which results in the halo nucleus breaking a p a rt into a core nucleus plus two neutrons. Since neither of the two body subsystem s in these two n eu tro n halo nuclei are bound against particle decay, th e forces which keep these nuclei bound m ust be due to the three body interaction of th e core nucleus plus th e two individual neutrons. Hence th e s tru c tu re of these nuclei is b est described by a three body system and they are often referred to as “Borrom ean" nuclei, due. 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(24) neutron drip-line. n. □. 1 n halo nuclei neutron unbound. 2n halo nuclei (Borrom ean). stab le nuclei. 2n halo nuclei. Neutron Number Figure 1.1. C h art o f th e nuclides in the region of p ro to n numbers from Z = 1 (Hydrogen) to Z = 6 (C arbon) showing the location of th e various types of neutron halo nuclei.. to the sim ilarity of th e ir stru ctu re with the B orrom ean rings which consist of three interlocking circles [4]. (A n example of such a com parison can be seen in Figure 1.2 where the B orrom ean rings are superimposed onto a schem atic diagram of th e three body stru ctu re of th e B orrom ean nucleus 11Li.) Because of this unique structure, Borromean nuclei ex h ib it properties which have n o t been observed before in th e nucleus. In order to pro p erly m odel and understand B orrom ean nuclei, it is necessary' to determ ine w hat th e stru c tu re of the two body subsystem s is within the nucleus. W hile the stru ctu re of th e neutron + neutron system is relatively well understood, the structure of th e core 4- neutron subsystem is q u ite u n certain for m ost of these. 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(25) Figure 1.2. Schematic diagram of the Borromean nucleus n Li.. Borromean nuclei. In the case of 11Li, which has been the m ost studied halo nucleus to date, th e stu d y of the core 4- n eutron subsystem involves a determ ination of the stru ctu re of th e unbound nucleus, 10Li [4], One m ajor question which exists concerning th e stru c tu re of l0Li is the location, spin and parity of the ground state configuration.. Previous studies on th e stru ctu re of 10Li have produced varying. results and do n o t provide a clear picture of the structure of 10Li. This is illustrated in Table 1.1 in which th e results of the previous studies of 10Li are listed in term s of the reaction used in the experim ent, th e energies and w idths of th e states th a t were observed as well as the assignm ent of th e sta te in the stru ctu re of 10Li. Of particular in terest is th e possible presence of a low-lying s-wave state th a t was first observed by K ryger, e t al. [5] who investigated th e neutron decay of 10Li from th e fragm entation of lsO an d observed a s ta te which decays by very low energy neutron emission. O th e r groups have also observed a sim ilar state which decays w ith low energy n eu tro n em ission [6, 7, 8, 9]. The presence of a low-lying s-wave s ta te in 10Li. 4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(26) Table 1.1. Sum m ation of the previous studies of the structure of I0Li ii. R eaction ,JBe(9Be.8B )10Li n B ( - - , p ) 10Li lsO + natC fragm entation n B(7Li.8B )10Li 11Li + C stripping 10Be(12C ,l2N )10Li 9Be(13C ,12N )10Li Breakup of 11Li 180 + 9Be fragm entation 9Be(9Be.8B )10Li. —S n (MeV) 0.80(25) 0.15(15) <0.15 <0.10 0.54(6) <0.05 0.24(6) 0.53(6) 0.21(5) 0.62(10) <0.05 0.50(6). T (MeV) 1.2(3) < 0.4 <0.23 0.36(2). S tate ground state ground state ground state ground state (s[/2) excited state (p i/2) ground state (sl/2) (Pi/2 ). Ref.. ground state (Pl/2 ). [G] [14]. 0.30(8) 0 .1 2 i° ‘° 0.6(1) 0.40(6). [11] [12] [5] [7] [8] [13] [13] [9]. has im portant theoretical consequences in the m odeling and understanding of 11Li as has been dem o n strated in the work of Thom pson, et al. [10] among others. In order to clarify the am biguity surrounding th e stru ctu re of 10Li in term s of the states th a t are present in 10Li. as well as to identify th e spins and parities of th e states, a stu d y of the s tru ctu re of l0Li was perform ed via th e 9L i(d,p)10Li reaction. It is the results of this stu d y which are presented in this work. T his thesis is organized into the following chapters. In C hapter 2, the principles of nuclear physics which are relevant to this work will be discussed together w ith various aspects of halo nuclei, specifically th e two neutron halo nucleus, 11Li, and how it is related to th e stu d y of the unbound nucleus 10Li. In C hapter 3, th e experim ental m ethod will be presented, and a discussion of the experim ental technique used to produce and stu d y the unbound nucleus 10Li will be given. In C h ap ter 4. a system atic analysis of the experim ental technique will be presented, along w ith the analysis of the l0Li d ata. In C hapter 5, th e results of the d a ta will be interpreted and com pared w ith other recent experim ents.. 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(27) CH A PTER 2 T H E O R E T IC A L BACKGROUND. 2.1. B inding Energy T h e binding energy of a nucleus is defined as the mass difference betw een a. nucleus AX , an d th e m ass of th e p ro to n s Z , and neutrons !V. which m ake u p the nucleus: B = [Zm p. N m n — (m ( AX ) — Z m e)]c2. (2-1). where m p, m n . an d m e are the m asses of th e proton, neutron and electron, respec tively. an d c is th e speed of light [15]. As is evident from this definition, th e binding energy of th e nucleus results from a conversion of a portion of th e m ass of th e con s titu e n t nucleons into energy. T his conversion of mass into energy is a result of the in teractio n betw een th e individual nucleons w ithin the nucleus. T h is in tera ctio n is governed by th e stro n g force which is a n a ttra c tiv e force between two nucleons when th ey are sep arated by approxim ately 2 - 3 fm, b u t turns repulsive if th e two nucle ons com e w ith in 0.8 fm of one an o th er [15]. Because the nucleon-nucleon force is a ttra c tiv e for only a sh o rt range of distances, any particular nucleon w ithin a given nucleus is only bound to those nucleons which are nearest to it. Hence, nucleons which exist on th e surface of a nucleus are in general less bound to th e nucleus th a n those nucleons in th e interior of th e nucleus since the surface nucleons have fewer neighbors w ith which to interact. A related q u an tity to the binding energy of the nucleus is th e sep aratio n energy for th e rem oval of a neutron or p ro to n from the. 6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(28) nucleus in question. T h e n eu tro n separation energy S n for th e nucleus. X is defined. as: S n = [m(A~lX ) + m n - m ( AX ) \ c 2. (2.2). A sm all neutron sep aratio n energy indicates th a t the last n eutron is weakly bound to th e nucleus w hereas a large neutron separation energy indicates th a t th e last n eu tro n is more tig h tly b o u n d to th e nucleus. In addition to its influence on th e binding of th e nucleus, the nucleon-nucleon interaction also strongly affects th e density of nuclear m a tte r w ithin th e nucleus. Because the nucleon-nucleon interaction turns repulsive w hen two nucleons are sep a ra te d by less th a n 1 fin or so, an average separation is m aintained between the nucleons within th e nucleus. T his implies th a t th e density of nuclear m a tte r re m ains relatively co n stan t w ithin th e interior of th e nucleus regardless of the num ber of nucleons th a t are in th e system . Since adding nucleons to th e surface of a nucleus subsequently decreases th e binding energy of th e nucleus, th e m ost tightly bound nuclei would be those w hich have a minimal am ount of surface area relative to their volume. Hence, th e m ost tig h tly bound nuclei are spherically sym m etric since the ratio of the surface area to volume for these nuclei are m inim ized [16]. S table nuclei are found in a narrow region of proton an d neu tro n num bers known as th e valley of stab ility where, for light nuclei (A < 40), th e num ber of neutrons is roughly equal to th e num ber of protons w ithin th e nucleus. Nuclei which deviate from this ratio of n eu tro n s to protons th a t is seen in th e valley of stability have decay lifetimes due to th e fact th a t (3 decay now becom es energetically favorable in order to increase th e sta b ility of th e nucleus. If th e ratio of neutrons to protons deviates further from th e ratio s seen in the stable nuclei of th a t m ass region, then particle emission becom es possible for the nucleus to decay tow ards a m ore bound configuration [16]. In th e case of th e Li isotopes, for instance, the addition of two 7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(29) n eutro n s to th e stable nucleus 7Li leads to the form ation of th e particle stable nucleus 9Li. A dding one more neutron to 9Li leads to the unbound nucleus l0Li which im m ediately decays upon form ation via neutron emission back to 9 Li. Since an ad d itio n al neutron cannot be added to 9Li to form a particle stable nucleus. 9Li is considered to exist a t the neutron drip line, since a neutron added to 9Li would ju st d rip off and n o t form a bound system . This would seem to indicate th a t 9Li is near th e lim it of stability for the neu tro n rich Li isotopes. W hile the addition of one n eu tro n to 9 Li does not produce a particle stable nucleus, th e addition of two neutrons to 9Li does yield th e particle stable nucleus 11 Li. T h e fact th a t 11Li exists as a particle stable nucleus indicates the presence of. certain p ro p erties and characteristics w ithin 11Li th a t are unique relative to those p roperties seen in nuclei near th e valley of stability.. One obvious characteristic. th a t is different from stable nuclei is the presence of the n eutron halo in 11Li. The basic principle behind the existence of a halo can be understood through the use of a p o ten tial model to describe th e interaction between th e core nucleus and the n eu tro n s in th e halo [3]. If th e separation energy for th e outer one or two neutrons in a nucleus is less th an 1 MeV, th e probability of their existence a t large distances relative to th e center of the nucleus increases dram atically. Assum ing the potential betw een th e core nucleus and a loosely bound neutron to be spherically symmetric, given by W (r), th e radial wave function is determ ined by the eigenvalue equation:. + 1 V ( r ) ) 0 = E,p. +. (2-3). where h is P lan ck ’s constant divided by 2w, n is the reduced m ass of the system. I is th e an g u lar m om entum of th e state, and E is the energy of the state. Setting tl'(r) = u { r ) / r so as to elim inate first-order derivatives, we get th e modified radial. 8. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(30) wave equation: ff-,2 h 2 cPu(r) h % l + 1) + W (r) + 2n d r 2 L M r 2). —. u(r) = Eu(r ). (2.4). T he sim plest in teraction between the n eu tro n and th e core nucleus can be rep resented by a square well potential of th e form:. W(r) =. w here R is the w idth of th e potential from th e center of th e nucleus. If we c onsider R to correspond to th e radius of the core nucleus, th e n th e behavior of th e wave function of th e n eu tro n a t large distances away from th e center of the nucleus will be determ ined in th e region where r > R. In th e sim plest case of an I = 0 bound n eu tro n (E < 0). th e wave function outside the p o ten tial is given by [3]: 2ire~Kr ekR ^ (r) “ 7T r (1 + k R ) 1/2 w here. k. (. *. is the wave num ber outside the potential: to ^ (2.6). K — \j an d k is the wave num ber inside the potential:. T h e density d istrib u tio n of th e neutron outside th e nucleus is given by: p(r). =. |^ ( r ) |2 47r2 e~2nr k?. e2kR r 2 (1 + k R ). ( 2 .8 ). As th e energy of th e s ta te approaches zero, which indicates th a t th e n eu tro n is becom ing more an d m ore weakly bound, the p aram eter. k. becomes sm aller w hich. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(31) in tu rn increases th e range of th e neu tro n density distribution [3]. T his increase in th e range of th e density d istrib u tio n can be seen as a tail in th e distribution w hich is referred to as the halo.. Such an increase in the d ensity d istribution is. illu strated in Figure 2.1 where th e form of the density d istrib u tio n for a neutron th a t is bound by 0.2 MeV (dashed curve) is shown relative to th e d ensity distribution for a n eu tro n which bound by 8 M eV (solid curve). In each of th e two cases, the density d istribution w^as determ ined assum ing a potential w ith a 2.6 fm w idth. The developm ent of the neu tro n halo is thus understood, in p art, as a result of the weak binding of th e last one or two neu tro n s in th e nucleus. In the case of 11 Li w'here the two n eu tro n separation energy is 0.247 MeV [3]. the range of th e last two neutrons in th e nucleus extends o u t q u ite far form ing the two neutron halo in 11Li. T h e effect th a t th e neutron halo has on th e density distribution of 11Li is q u ite rem arkable considering the fact th a t th e root-m ean-square radius Rrms of th e n eu tro n halo is 4.8 ± 0.8 fm [3] which is nearly th e sam e size as the Rrms of 208P b of 5.5 fm [17] an d twuce th e size of 7Li w ith RrmS = 2.33 ± 0.02 fin [18]. W hile th e nature of the n e u tro n halo can be understood in term s of a simple p o ten tia l model, m any other unique facets of 11Li cannot be described in such a sim ple m anner. One such feature w hich requires a more com plete description of the s tru c tu re of 11 Li is the three b o d y interaction between th e two valence neutrons in th e halo and the 9Li core nucleus which causes 11Li to rem ain bound against particle decay. To construct a com plete description of 11Li. all of the possible interactions betw een th e 9Li core an d the two valence neutrons m ust be accounted for within th e calculation of th e wave function for th e system. The wave function for the two n e u tro n halo nucleus can be d eterm ined using the eigenvalue equation: Htp = E 0. 10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.. (2.9).

(32) 10 E = 0.2 MeV E = 8 MeV. 10. 10. 10 ". 10,.4. 10'■50. 4. 6. 8. 10. r (fm ). Figure 2.1. N eutron density distribution for a neutron bound by 0.2 MeV' (dashed curve) and 8.0 MeV (solid curve) based on the square well potential m odel.. where H is th e H am iltonian for th e three body system which has the form [19]: H = E W _ + Pi2l + Vne{1) + Vnc{2) + y nn + (P (l)+ P ( 2))~ Zmn Zmn ZAcTrin. (2 10). T he first two term s in the H am iltonian are the kinetic energies for the two valence neutrons, labeled (1) and (2), Vnc is th e interaction of a specific neutron w ith th e core, Vnn is th e neutron-neutron interaction, and th e last term is the kinetic energy of th e recoiling core nucleus which has the mass num ber A c. As seen from the form of th e three body H am iltonian, an im portant term in developing a com plete under stan d in g of 11Li is the interaction between the core nucleus and a valence neutron, which is determ ined by the stru ctu re of the unbound nucleus, 10Li. Hence, an un derstanding of the stru ctu re of 10Li is essential in developing a better understanding of 11 Li. One can deduce some properties of 10Li based on an understanding of th e shell stru ctu re for th e nucleus. 11. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(33) 2.2. Shell Model The basic principle behind the nuclear shell model is to describe th e various. energy states w ithin the nucleus based on the nucleon degrees of freedom [16]. Each nucleon occupies a specific single particle level w ithin the nucleus which can be cal culated using a potential model to describe th e binding of the individual nucleon w ithin the nucleus. Choosing a realistic potential to represent th e interaction be tween th e valence nucleons and th e core nucleus is crucial in o rder to produce an accurate m odel of the nucleus. Two simple forms th a t can be used for this poten tial would be the square well potential as described in the previous section, and the three dim ensional harm onic oscillator. However, neither p o ten tial reproduces the properties of the nuclear potential accurately due to discrepancies between the edge of these potentials and the edge of the surface of the nuclear m a tte r distribu tion. which closely resembles the edge of the nuclear potential [15]. One form th a t resembles the nuclear potential more closely is the Woods-Saxon potential:. V{ r ) =. 1 -F e. “. <2 1 1 >. where R is th e m ean radius of the nucleus, and a is the skin thickness which is defined as th e distance over which th e potential changes from 0.9K, to 0.lV'o [15]. To increase th e accuracy of this shell model in predicting th e ap p ro p riate be havior of the nucleus, an additional potential is added to account for th e spin-orbit interaction which couples the intrinsic spin of the nucleon, s, w ith its angular mo m entum . /. an d takes the general form: Vso = V ( r ) r - s. (2.12). where V (r) is th e radially dependent stren g th of the spin-orbit term which is often determ ined by fitting the observed single particle levels in the given nucleus [16]. 12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(34) Figure 2.2. Level structure in 10Li based on naive shell model.. A lthough sim ple in construction, this naive shell m odel can be useful in devel oping a starting p o in t for attem p ts to u n d erstan d th e basic properties of specific nuclei. Figure 2.2 shows th e filled energy levels for th e p roton and neutron shells for 10Li based on this naive shell model. As seen in Fig. 2.2. 10Li has one proton ( tt) in the lp 3/2 shell and one neutron (u) in th e l p i /2 shell. T h e ground state configuration for 10Li in this naive shell model is thus determ ined by th e interaction between these two valence nucleons since th e rest of th e nucleons com pletely fill the lower energy levels and are thus assum ed to form an inert core w ith 0+ spin and parity. In th is configuration then, th ere are two possibilities for th e ground s ta te spin and p arity of 10Li depending on how th e total angular m om entum of th e valence proton an d neu tro n couple together: [ 7 r lp 3 / 2 ® l d p i / 2 ]. =» 1+. (jl. =>. (ji+h). 2+. -. j 2). To check if this ty p e of assignment for th e ground s ta te of 10Li is accurate, one can apply this same m odel to other nuclei having th e sam e num ber of neutrons as 10Li. In the case of 11Be, w hich has one more p roton th a n 10Li an d has an experim entally 13. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(35) d eterm in ed ground s ta te spin and p a rity of l / 2 + [20 ], the application of th e naive shell m odel to l l Be would yield a l /2 ~ assignm ent for the ground s ta te due to the presence o f th e lone valence n eu tro n in th e lp i /2 shell. This discrepancy in the p a rity of th e ground state of 11Be betw een the naive shell model an d the experim ent indicates th a t th e level stru ctu re of 11Be does not follow th e predictions of the naive shell model, b u t rather has a configuration where the 2 s^/o shell is lower in energy th a n th e lp i /2 shell. T his discrepancy between the m easured shell stru ctu re of 11 Be h as im plications for th e stru c tu re of l0 Li. In calculations which were first perform ed by Sagawa, et al. [21], th e inversion of the 2 si / 2 shell w ith th e lp i /2 shell was reproduced by considering th e coupling of an excited core nucleus in 11 Be to n eu tro n s in th e sd shell. These sam e calculations also predicted a possible inversion of th e l p i /2 an d 2 si / 2 states in 9He, as shown in Figure 2.3[22]. Since b o th 11 Be and 9He are N = 7 isotones, this implies th a t 10Li m ay also have an inversion between th e l p i /2 a n d 2 s i / 2 levels. Such an inversion would have strong im plications for the gro u n d s ta te configuration of 10Li, w ith th e valence proton in 10Li now coupling with a 2 s 1/2 n eu tro n yielding a ground s ta te configuration of: [tt1P3/2 ® ^ 2 s i/2] =► 1“. (ji - j 2). I 11 a d d itio n to this possible change in th e parity of the ground s ta te configuration for 10Li from th e naive shell model, th e calculation also suggests th a t th e 2 s i / 2 state would lie ju s t above th e threshold energy for particle decay in 10Li. Because this is an s-wave n eu tro n state, it would be possible for the s ta te to overlap th e threshold due to th e fact th a t it is not affected by th e presence of the C oulom b b arrier or the centrifugal b arrier of th e 9Li nucleus. If th e position and w idth for such a s ta te were such as to overlap th e threshold, th en th e s ta te would be ‘v irtu a l 1 since it is neither. 14. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(36) 9H e. 10Li. 11Be 12B. 13C. 14N. 150. 0 --------------. -1. -. \. -2 . 3. 4. ' I E* - S n [MeV] -. . .. —. .. N = 7 iso to n es. -5 ■ *B.A. Browr (1996) E 1/2- - E!/2«- 5 0 M . 9 MeV. Figure 2.3. Plot of th e calculated difference between the excitation energy of th e l / 2 + and 1 /2 “ states an d th e neutron separation energy for N =7 isotones (from Ref. [22]).. com pletely a bound s ta te nor is it a resonant state. T h e concept of a virtual s ta te will be further discussed in th e next section 2.3. Scattering Theory T h e most common technique for determ ining th e properties and stru ctu re of. various nuclei is to perform experim ents where an incident nucleus is scattered from a ta rg e t nucleus in order to form specific final states. To understand th e results of such scattering experim ents, it is first necessary' to u nderstand the scattering process by which the desired s ta te is formed. In a typical nuclear scatterin g experiment, a beam of particles with m ass M { and w ith an energy, E 0, is incident on a target nucleus w ith m ass A/ 2 which is usually a t rest in the lab. T he wave function of the com posite scattered system, ip(r), is. 15. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(37) determ ined by: O -hV 2ifr(r) -+- V(r)ip(r) = Eil>(f). (2-13). where E is th e energy of the relative m otion of th e two particles which is related to the incident energy E a by [23]:. £ = a. (214). and the vector f is th e relative position between the two nuclei. T h e incident wave can be represented by a plane wave of th e form: ibiir) = A e , k f. (2.15). where k is re la te d to th e initial m om entum of th e incident nucleus by:. E- f. <216). and A is a n o rm alizatio n constant for th e wave function. U pon scatterin g , th e wave function moves radially outw ard and takes the form: e ik r. t i / ( r l = / ( M ) ----r. (2.17). where f( 6 , <p) is a n angular function which describes the dependence of th e scattering on the angle of th e outgoing nucleus. If the sc atte rin g potential is of finite range, then a t large sep aratio n s th e wave equation reduces to one for a free p article which implies th a t the wave function m ust satisfy the asy m p to tic boundary condition [24]: pikr T P ( r ) ~ A { e ' k-r +f{6,4>) ---- ) r. (2.18). T he differential cross section would be given by [24]: ^. = l/(9 .* ) |2 16. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.. (2.19).

(38) If th e potential is spherically sym m etric and th u s dep en d en t only on th e absolute distance between the two nuclei, then th e angular function can be expanded in term s of spherical harm onics YlTn{6, o) which leads directly into th e partial wave expansion of th e wave function: t/;(r) = f \ aim^ Y r { 0 , < t > ) r. ( 2 .2 0 ). 1=0. w here u /(r) satisfies th e m odified radial wave equation(E qn. 2.4)[16]. T he fact th a t th e scatterin g potential chosen in this case is spherical im plies th a t a separate radial wave equ atio n exists for each partial wave. At th e asy m p to tic boundary. ut m ust take th e form: Ul(r) = C /(e -<(fcr"5,,r) - e*Sie '( *r- & )). ( 2 .2 1 ). w here C[ is a constant which is determ ined by th e b o u n d ary conditions, and Si is th e phase shift of th e wave w ith singular m om entum I [16]. In th e case of elastic scattering, th e outgoing waves can only change in phase relative to th e incoming wave, and this is represented by th e phase shift S{. In th is case, th e angular factor f(Q, o) and subsequently th e differential cross section, is given in term s of th e phase shifts of the partial waves: da. 47r. dn =. ^ 2 V 2 f T l e tSl sin(J,)y]°( 0 ). ( 2 . 22 ). 1=0. w here it has been assum ed th a t the incident wave is traveling along th e z axis subsequently removing th e dependency of the sc atte rin g on the azim uthal angle. T h e elastic cross section is th e n given by: OC. 7T ° - = p H ( 2 ' + l) |l- S L ,[ * 1=0. (2-23). w here S ln Q is the elastic scatte rin g S-m atrix elem ent for th e partial wave w ith a n g ular m om entum I [25]. T h e elem ents of the S-m atrix S aQ describe the am ount of 17. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission..

(39) the incident wave function th a t is in th e elastic channel relative to th e am ount th a t was sc atte red into other open reaction channels. In this case. SQQ is given by: SL.a = e 2" 1. (2-24). T h e elastic scattering S-m atrix is useful in developing an understanding in the presence of v irtu al states in the scatterin g process as well as differentiating between a virtual s ta te and a resonance. C onsider an s-wave neutron incident on a nucleus w ith energy E . A resonance will occur if th e phases of the wave functions in the regions o utside and inside the nucleus m atches sm oothly at the boundary' r = a. T he S m a trix elem ent in this case can be expressed in term s of th e wave num ber of the incident n eutron by [25, 26]:. =. ( 2 ' 2 5 ). where u{r) is th e reduced radial wave function given in 2.21 and / is th e derivative of th e radial wave function a t the b o u n d ary given by:. A sim ilar relatio n can be m ade betw een th e S -m atrix element and th e energy of the incident n eu tro n from Equation 2.25: -2ikgE 2Z^° E — Er + i i r o where T 0 is defined as the w idth of th e resonance. The S-m atrix will possess poles in th e com plex energy plane corresponding to the resonances in th e reaction. For a single isolated resonance th a t is far from threshold, the location of th e pole in term s of energy' is determ ined by th e equation [27]:. E p o le. = Er - ^ 18. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.. (2.28).

(40) W hile this equation works for s-wave resonances which are well away from threshold, for those resonances which exist near threshold, such as th e possible unbound s-wave s ta te in 10Li. the energy dependence of the w idth due to presence of the potential barrier has an influence on th e location of the pole in th e complex k-plane for s-wave neutrons. A generalized condition for th e energy of th e pole which takes into account the probability of a neutron w ith angular m om entum I to p en etrate th e barrier of the nucleus is given in term s of th e energy-dependent w idth T(E) [27]: T ( E) =. r/0. P ( E ). (2.29). P t(E r)\. i E pole = ET - - T { E ). (2.30). w here Pi {E) is the p en etrab ility factor which is determ ined, in general, by using the coulomb wave functions Fi and Gi evaluated a t the channel radius R [28]: kR Pl ( E) = ( Fi ( kR))2 ( G i ( k R ))2. ( 2 '3 1 ). For a neutron w ith I = 0 or / = 1 [27]: P0( E). =. P‘(E) -. kR. (2.32). <TT& j. (2'33>. Because of the difference betw een the penetrability factors for s-waves and. p-waves.. th e two states behave differently near threshold. In th e case of th e p-wave. the pres ence of th e centrifugal barrier forces the cross section to go to zero as E approaches threshold. The cross section for a single resonance is given by th e Breit-W igner one-level formula:. (r(£))2 ^. -. k2 (E. -. Er )2 +. 19. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.. [. }.

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