Radhi, R A

Taqi, Ali H

IACS Institutional Repository

,^^^,,.„,xm.v.75A (3), 255-258 (2001)U Pan international lournuiRandom phase approximation with high orbits configuration for the low lyingnegative parity, T = 0 states in “’OR A Radhi*, Ali H Taqi** and A(K Ahmad***Dcparlmcnl of Physics, College iof vSciencc.University of Baghdad, Baghdad, Iratj Department of Physics, Collegcjol Science,Saddam University, Baghdad IraqP- mai I sadsc i (?’ ii ru k 11 n k»nct R e c e iv e d  1 0  M a x  2 0 0 0 , a c i c p t e d  1 6  O i t o h e t  2 0 0 0Vh t^Iact Eleclroexcilations of the dominantly low-lymg 7 = 0, negative paiity, paiticlc-hole states of '%) are studied in the framework i)f theuFiJtmi phase approximation (RPA) These states include stales with /" (E MeV) 1 (7 12).  ^ (6 H) and 4 (17 79) All possible single T  ~  0 pailicle- ii'k Slates of allowed angular momenta, for both ground stales and excited slates are considered in bases including single paiticle stales up to the II P h^l.'ll The Hamiltonian is diagonalized in this extended space in the presence of the modified surface delta interaction (MSDI) Admixture of higher . <i)!iniram>iu is considered foi the ground state to allow for a large scale of a collective motion in the nucleus fTfective operators are used to aLCounl ! I ilic coTc polarization effect The form factors calculated in this paper are m good agreement with the experimental dataknwards Nuclear reactions, form factois, random phase approximationl’\(\S Nos. . 25 30 Dh, 21 60 Ev. 27 20.4ii1. Introductionflic  M t n p l e s t  coni'iguralion for closed shell nuclei such as >i IS doubly closed shell. If such a system is excited through ihc s i n g l e  application of a one body number-conserving operator, then the resulting states must have a particle-holc pail The excited collective oscillation can be described as a linear combination of particic-holc states. Such an approximationcalled Tamm-Dancoff approximation (TDA) | l | .  In this approximation, there is an asymmetry in the way the ground t^aic IS treated on one hand and the excited states on the other, ffic ground state is described in terms of one configuration ^^nlv and this is in fact corresponds to independent-particle ‘nodel. A  system of states more general than that considered in the TDA  appears when treating the ground state and the excited elates more symmetrically. In that case, one allows both to have particle-hole pairs. Such approximation is referred to as random- Phasc approximation (RPA) [I], The theory of the RPA is ^andamental to the collective motion in nuclei.The RPA model can be tested in two domains, first by calculating the energy eigenvalues and eigen vectors ^ turesponding Author.(amplitudes) through the diagonali/alion of the Hamiltonian, and second by comparing the calculated and measured electron .scattering form factors. In this paper, we study the isoscalai tremsition in which connccls the (7^ = 0^ ,7' = 0) ground state with the low-lying ; l~, 3" and 4 isoscalai stales. 1'hc Hamiltonian is diagonah/cd in the orbits Lv, I/?, 2,v, IzA 2/; and 1/ in the presence of the modified surface delta interaction (MSDDI21.In the calculafion of the form factor, the ground stale is modified to include admixture from higher configurations iiplo the orbits 3/>-2A A comparison of the electron scattering form factors calculated by using RF^ A model with the available experimental data for the low lying isoscalar, negative parity slates arc discussed.2. Theoi^2. y Random phase approximation (RPA) :In the RPA, the ground stiUc and the excited state are treated symmetrically, allowing both to have particle-hole pairs. This means that the excited stale can be reached either by creating or destroying particle-hole pairs in the ground state. The particles©2001 lACS256 R A Radhi, A ll H  Taqi and  A K  A hm adand holes ai*c labeled by a and b, respectively, each with quantum numbers (w/y>n l/2 /Jw ith r ,=  1 / 2  ( - 1 /2 ) for a proton (a neutron).The equations of motion are linearized and lake the form 111:where• ( la )a!/V  L 4 y ~ y 1 ~ i:-2Ld\, ^ u h , a ' h ' ' a ' h '  ^ a h . a ' h ' ^  u ' h ' \  *ah (lb)^uhM 'h ' ~  ~  ) ^uhM 'h ' K h ,a h '  » ®The matrix elements for particle hole states coupledtoy and Tare given by a sum of parlicle-particle matrix elements {a*h\V\ah'), coupled to different values of J' and 7 ' [2)= - S ( 2 V '+ l ) ( 2 r + l ) j ^ "j y  [ J u '  J hf i / 2 i / 2 r ] ,l / 2 1 / 2 r PThe panicle-particle residual interaction used in this work, is the modified surface della interaction (MSDI) [2].The coefficients X  and Y are the amplitudes that describe the creation or destruction of particle-hole pairs in the ground states, respectively. The quantities are the unperturbedenergy of the particle-hole pair and e  are the energy eigen values for the different excited slates of the given J T  values. Eq. (la) and (lb) can be written in matrix form as( A B \BAX  \ 0 \  y ' ~ ^ 0 - 1X(5)and the diagonalization of it will give the eigen values e and the eigen vectors X and K. The states of different spin-isospin (JT) values are decoupled by these calculations.Once the secular matrix (eq. (5)) has been diagonalized and the amplitudes have been obtained, the matrix elements of the required multipolc operators given in terms of the single-particle matrix elements by [ 1 )( n r |  f / r |  ‘Hoo) = X [ ( « | | |J Tah (6)where the triple bar denotes a reduced matrix clement with respect to both angular momentum and isospin.The single-particle matrix element reduced in both spin  ^isospin, is written in tenns of the single-particle matrix elcrncn reduced in spin only [3Jwith ( ~ 1) M 2 ~ t ,f o r  7  =  0  f o r T ^ \ '2.2 Electron scattering form factors :Electron scattering form factor involving angular moincnuim and momentum transfer 7^ , between the initial and final nuck; shell model states of spin 7^  ^and isospin 7^ are [3 |4 k T, TT,Z- ( 2 J , + \ )  r.oA - r o i :with 7] selecting the longitudinal (L), transverse electric i /i) ai transverse magnetic (M) form factors, respectively. / tl projection of the initial and final states and i s  g i \ c i i  t T  = (Z ~ A ()/2 . The finite size (f.s) nucleon form (.lUoiFf f q )  = cxp(">0.43«/^ / 4) and F^ = exp(r/‘/>' /44i the correction for the lack of translational invariance in the six model. A  is the mass number, and h  is the harmonic oscilkii' size parameter.The total longitudinal (7) and transverse (7) form laciors a given by|2/iO(Ky>oThe single-particle matrix elements (^ |^|^/!. ||^ )^ **required electron scattering operators used in this work are lln> of Brown etal. [4).3. Results and discussionElectro-excitation of the low-lying isoscalar, negative pan states in are tested and compared with the availab  experimental data [5,6J, in the framework of RPA with imnl space including all orbits up to the 2p-l/shcll. The form lacto are calculated with ground stale wave function which is modiia to include higher configurations upto the orbits of 3 /> -2 /  she using mixing parameter y that mixes the state \nlj) with state The isoscalar states that arc calculated in ihwork include states with (E M eV ): 1” (7.12), 3" (6.13) and (17.79).R andom  phase approxim ation with h i^h  orbits conftgurarion etc 257In ihc simple shell model calculations, the ground state of assumed to form closed *P)/2 shells.\wording to RPA, the ground states as well as tlie excited ' [dies arc treated on the same fooling, and all possible ,, l^iauralions for the ground slate and the excited states are .^nsiruLicd by removing a particle from the closed shells and loinoling It to higher shells leaving a hole state within the \)scd shells. Experimentally, the slates 1 “, 3' and 4~ are found ji7 1 2  6.13 and 17.79, respectively [5]. Our RPA calculations .‘edict the values 8.34 MeV, 10.83 MeV and 17.19 MeV,c^^ pcctlvely.The longitudinal form factor for the 1 (7.12 MeV) state is diDwn in Figure I as a solid curve for comparison with the c\[>cninental data. The data do not show any diffraction minimum and cannot be reproduced through all the momentum transfer \.ducs. Same observations were made by Vincent and Vinh ManJ =1". Tn-O, .83 fm and E,r=7.12 MeVHj»iire 1. Longitudinal form factor for the 1 spurious stale17J. RPA calculation based on quantum hydrodynamics which IS commonly referred to as mean field theory (MFT) [8] predicted that the first stale is a spurious state and gives a spurious excitation energy of 0.5/ MeV which is indeed zero within the numerical accuracy of their calculation. The RPA based on lelativislic Hartree approximation (RHA), also predicted a spurious state but with higher energy than that of MFT [8].calculation predicts the second state which is the non* "'Purious slate at 8.47 MeV, while the RHA calculation predicts this state at around 13 MeV. Their result for this state, agrees in "•hape with longitudinal form factor data, but underestimates In our work, the non-spurious state is at 15.58 MeV, which ‘inclose to the RHA results [8], The calculated longitudinal form for this state is shown in Figure 2 and compared with the ^^perimental data [5], where the data are fitted for medium range ’^1 q only. In our calculation, we use effective charge equal to and 0.45e for the proton and neutron, respectively, to account for the core polarization effect for the longitudinal formfactor (9|. Also admixture of higher orbits are taken into consideration with y equal to 0.9. The viiluc of the size parameter h of the HO potential for the single particle wave function used in this state is 1.83 fm which is consistent with that of Ref. 151 The deviation of the calculated energy levels and the longitudinal Ibnn factor for this stale reHecls the fact that enormous degrees of collectivity are required lor proper treatment. The present as well as all previous results reveal that this Slate needs moic investigation from the theoretical point of view.’®0. J =1", T=0. b 83 and E =7 12 MeV0 0 0 5 2 0 2 5 3 010 15q(fm)^Figure 2. Longitudinal form factor for the I non-spunou.s .stateThe 3 (6.13 MeV) slate is calculated to be equal to 10.83 MeV. Our RPA result is consistent with the RHA-RPA results of Rcl*. 181 (~ 10 MeV), where their MFT result is 5.99 MeV, which is clo.se to the experimental value. Our RPA calculations for the longitudinal foim factor is shown in Figure 3, which is based on’»0, J rr3-, T::rO, b^ =1.8 fm and E =6 13 MeVF ig u r e  3. L o n g itu d in a l fo rm  fa c to r  fo r  th e  3 (6 .1 3  M eV ) sta te .258free-nucleon  g-facto rs. C o re  p o la riza tio n  effect is takenR A Radhi, Alt H Taqi and A K  Ahmadthe  s ing le -p artic le  w ave  functions o f  the  H O  po ten tia l w ith  size param ete r 1.8 fm  and  w ith  effec tive  charges Cp = 1 ,1 e and =  0 .1 5e, fo r the p ro ton  and neu tron , respectively . A dm ix tu re  o f  h igh er o rb its  in the  g ro u n d  slate fo r the  3 '  ( 6 .13 M eV ) sta te  is less im portan t from  tha t in the I " (7 .12  M eV ) sta te  and  needs sm all e ffec tive  charge  to  d esc ribe  the  data. O u r resu lt fo r this s la te  a g re e s  in  s h a p e  a n d  m a g n itu d e  q u ite  w ell w ith  the  experim ental da ta  for all m om entum  transfer values.O ur last exam ple  is the 4‘ (17 .79  M eV ) stale, w here only the transverse  M 4 m u llipo le  co n trib u tes to the scattering . O u r RPA re su lt p red ic ts  the  lo w e s t 4 '  .slate a t 17.19 M eV, w h ich  is consistent w ith the m easured value at 17.79 McV. O ur calculation for the  form  fac to r is show n in F igure  4  as a  dashed  curve. T he ca lcu la tion  in co rp o ra tes the sing le  partic le  w ave function o f  the H O  po ten tia l w ith /> =  1.76 fm . T he  ca lcu la tion  ag rees in shape,' * 0 ,  J  =4-,  T x O ,  b = 1 .76  Im and £ , = 1 7 . 7 9  M e V  1 E - 3  , " r'i ■ .|."|. r  i. I t I i yr-r-T-1—|-r-T-r-r—  ■ OurRPA(g=0.55v=09) OurRPA(g=055vs10)------- OurRPA(9=10v=10)IE -4  •  C E Hydetal(1987)accoun t to  sup p ress the  fo rm  fac to r as show n by the cu rv e  in F igure  4. A dm ix tu re  o f  h ig h e r configuration arc included  in the  g round  sta te  w ave func tion  w ith y = 0 y is very im po rtan t to  add  m ore  deg ree  o f  collectivity . An ove, ,| agreem ent is ob ta ined  w ith  the experim ental data with g lactur -0 .55  and y =  0 .9 , as show n by the  so lid  cu rv e  in Figure 44. ConclusionsW hen the .space o f  w ave  fu nc tio ns is ex ten ded  to  include or|,n up to  2 p - l f  shells, RPA  resu lts  g ive reasonab le  dcscripiKw i the da ta  fo r the e lec tron  sca tte ring  resu lts. A n improvenicni ob ta ined  w hen  w e a llow  h ig h er she lls  up to  the 3p-2f shell A lso  inc lud ing  co re  po la riza tio n  e ffec ts  through  the eticcn\ charge  m odel fo r the  long itud ina l sca tte ring  and effcciivc > fac to rs  fo r the  tran sv erse  sca tte ring , im prove the agiecinci w ith  the ex perim en ta l d a ta  and  gave good  description im ,i a v a ila b le  m o m e n tu m  t ra n s fe r  d a ta . S u c h  ingrcd icnis a: necessary  in the m odel to acco u n t fo r a large scale o f cnllaH' m otion in nuclei.R eferencesFigure 4. Transv ers e magnetic fo r m  factor for the 4  ( 1 7 . 7 9  M e V )  statebu t ov erestim a tes  the  da ta  by  a fac to r o f  -  10. Sam e sort o f  resu lts  is o b ta ined  by R H A  [10]. C o re  po la rizatio n  ca lcu la tion s [11,12] yield values for the g-factors reduced from  the free values, w hich arc called effective g-factors. E ffective M 1 g-faclors equal to 0 .8  o f  the  free  nu cleon  va lues are  found  adeq uate  to describe  the m agnetic  m om en t and  the e lec tron  .scattering data, w hile 0 .6  o f  the free  n u cleon  g -fac to rs  arc needed  to desc ribe  the M 3 e lec tron  sca tte ring  d a ta  [12]. T h e  effec tive  g -fac lo rs  used in th is  w ork  to  desc rib e  the M 4 fo rm  fac to r arc  equal to  0 .55  o f  the12]l-^ I[41l.SlW171[81[911101i m112]T  de Forc st  and J I )  Walccka Adv PIm  1 5  I (1966)P J Brussaard and P  W  M  Gla ud cm a ns  Shell Modil Ainflnuuo in Nuclear Spectrm npv  (A ii is ie rd a m  . N o r t h  Holaiui) ih)7"W  T  D o n n e l l y  and I Sick  Rev Mod Phys 56 461B  A  B r o w n ,  B  H  Wildenthal,  C  F  Willia mson , F  N  Rad. S Kuw;iK H  C ro nn el  and J T  O ' B r i e n  Pfivs Rev C 3 2  1 1 2 7  (19HSiT  N  B ut i,  J  J Ke lly ,  W  Be rt oz / J,  J M  F i n n ,  F  W  Hcrsrnan. (> H\i M  V  H yn e s , M  A  Kova.<»h, S K o w a ls k i.  R  W  L o u n c  H L  Noiu B P u g h ,  C  P Sargent  and W  T u rc h i n c s ts  Phys Re\ '( 1 9 8 6 )C  E  H y d e - W n g h l .  W  B c r to / z i ,  T  N  Bu ll ,  J M  Fi n n , F  W  llcrsm M  V  llyncs,  M  A  K o v a s h , J  J  K e ll y ,  S Ko w a ls k i,  R W  Uuiiic I N o r u m ,  B  Pu gh,  C P  wSaigcnt. B L Berman . F  Petrov icIt .mil J C a r r  m . v  Rev. C 3 5  880 ( 1 9 8 7 )G  Vincent and N  V i n h  M a u  Nud. Phys 54 3 1 2 ( I % 4 )C  E  Price. E  Rose. J R  Shepard and J A  M c N e i l  Ph\s Rc\ ( 10 89 ( 1 9 9 2 )B  A  B r o w n ,  R  R a dhi  and B  H  W il d e n th a l Phys Rep 101  ^( 1 9 8 3 )J Pi ck a re w ic z Nucl Phys. A 5 1 1  4 8 7  (19 90)T  S u z u k i ,  H  H y u g a ,  A  A r i m a  and K  Y a z a k i Nucl. Phys A358  ^( 1 9 8 1 )B  A  B r o w n ,  R  R a d h i  and B  H  W i l d e n t h a l  Phyx. i e i i  t . ^ «(1983)

Random Phase Approximation with High Orbits Configuration for the Low Lying Negative Parity, T = 0 States in 16O

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Random Phase Approximation with High Orbits Configuration for the Low Lying Negative Parity, T = 0 States in 16O

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