III. Effective Interaction Theory

Transcription

1 III. Effecve Ineracon heory opcs o be covered nclude: Inuve deas Bref revew of operaor formalsm ason scaerng formalsm Kerman, McManus, haler scaerng formalsm Feshbach scaerng formalsm Brueckner nuclear maer formalsm Relavsc mulple scaerng eneral References: Ray, Hoffmann, Coker, hyscs Repors 22, Rodberg and haler, Inroducon o he quanum heory of scaerng, cademc ress, 967.

2 he nuve deas ha led o effecve neracon heory he basc deas for dealng wh he many-body, srong non-perurbave nuclear neracon problem began wh scaerng so ha s where I wll sar. he semnal dea was due o Lesle L. Foldy who as a recen hyscs B.S. graduae n 94 was workng on sonar durng II n New York Cy. Buldng on hs experences wh acouscal waves Foldy n 945, us before enerng graduae school o work wh J. R. Oppenhemer a U.C. Berkeley, and afer hs work was declassfed, publshed he landmark paper L. L. Foldy, hys. Rev. 67, Foldy descrbed proecle scaerng from a nucleus as a wave propagang hrough many, dense scaerng sources wh a complex absorpve ndex of refracon. Hs essenal dea was o express he oal scaered wave n erms of ndvdual N+N scaered waves, raher han n erms of he very srong N+N neracon whch canno be expanded n a perurbaon seres, and may even dverge n he case of hard-core N-N neracons. nce rbue o Foldy s: hp://sng.phys.cwru.edu/pl/apersindf/206foldy.pdf Hs 945 heory of he mulple scaerng of waves lad ou he fundamenals ha mos modern heores have followed and somemes redscovered, No bad for a pre-graduae suden! 2

3 he nuve deas ha led o effecve neracon heory In 950 eoffrey Chew nroduced he mpulse approxmaon as a suable way o smplfy he nracable + body problem e.g. p +, n + o an effecve wo-body scaerng problem. hree papers esablshed he basc deas for wha would become known as Mulple Scaerng heory: Chew, hys. Rev. 80, Chew and ck, hys. Rev. 85, Chew and oldberger, hys. Rev. 87, he bascs are: he full + scaerng can be accuraely represened as a coheren sum of ndvdual hadron + nucleon scaerngs and re-scaerngs from nucleons n he arge nucleus 2 a hgh energes he free-space hadron + nucleon scaerng amplude s unaffeced by he nuclear medum 3 he hadron + nucleus scaerng amplude should be expanded n erms of he wo-body scaerng ampludes, raher han drecly n erms of he N+N poenal. In 95 Melvn Lax Rev. Mod. hys. 23, , hys. Rev. 85, exended hese approaches o oban an effecve neracon poenal, laer called he opcal poenal o represen he effecve p + neracon. hs was he frs represenaon of such an effecve poenal and nroduced he so-called r form, where r s he nuclear densy and s an effecve N+N neracon. Hans Behe learned abou hs and n a semnar passed hs new dea on o Roy haler my 2 nd menor n he lae 70 s and 80 s who devoed hs career o mulple scaerng formalsm. 3

4 he nuve deas ha led o effecve neracon heory In 953 Kenneh M. ason gahered up all hese emergng deas and publshed he frs, formal scaerng soluon for he p + problem n K. M. ason, hys. Rev. 89, Hs heory wll be presened n laer sldes. K. M. ason In 959 rhur Kerman, Hugh McManus and Roy haler correced a double counng problem n he ason heory by re-organzng he expansons nn. hys. 8, whch paved he way for accurae applcaon of Chew s mpulse approxmaon and led o many applcaons for scaerng expermens. K M 4

5 he nuve deas ha led o effecve neracon heory Herman Feshbach and collaboraors, a he same me, developed a powerful proecon operaor formalsm whch hey used o generae a perurbaon expanson of he opcal poenal and whch could be appled o reacons oher han elasc scaerng [nn. Rev. Nucl. Sc 8, ; nn. hys. NY 5, ; nn. hys. NY 9, ]. Keh Brueckner he success of mulple scaerng formalsm and he effecve neracons was noced by nuclear srucure heorss. hese deas were ncorporaed no heores of nfne nuclear maer by Keh Brueckner and Levnson n 955 hys. Rev. 97, Infne nuclear maer no Coulomb, no surface, no symmery energes o worry abou was a frs sep on he way o a heory of nuclear srucure. he resulng effecve neracons, called g-marx n he leraure, s ubquous n nuclear srucure calculaons. 5

6 Inroducon o Operaor Formalsm I assume ha mos of hs noaon s famlar o you and here offer only a quck revew of hose expresson whch are mos useful n nuclear scaerng and whch we wll need o undersand he effecve neracon formalsm. he followng are coped from Rodberg and haler, Ch. 6. 6

7 Inroducon o Operaor Formalsm 7

8 Inroducon o Operaor Formalsm 8

9 Inroducon o Operaor Formalsm 9

10 Inroducon o Operaor Formalsm Local oenal Schrodnger equaon n 0 operaor form

11 Inroducon o Operaor Formalsm

12 Inroducon o Operaor Formalsm 2

13 Inroducon o Operaor Formalsm rncpal value : f x f x f x lm 0 dx dx x x x Sae vecor n operaor form 3

14 Inroducon o Operaor Formalsm Moller operaor 4

15 Inroducon o Operaor Formalsm Defnon of -marx 5

16 Inroducon o Operaor Formalsm Summary: V V V V 6

17 ason Scaerng Formalsm Based on ason, hys. Rev. 89, wh updaed noaon 7

18 ason Scaerng Formalsm 8

19 ason Scaerng Formalsm Inracable, non-perurbave 9

20 ason Scaerng Formalsm hs auxlary -marx s readly approxmaed n erms of he free N+N scaerng amplude Chew s mpulse approxmaon. erurbave expansons can be consruced n whch many-body correcons are ncluded. see sldes for deals 20

21 ason Scaerng Formalsm see sldes for deals erurbave expanson of he opcal poenal n erms of nuclear correlaons dd/subrac elasc channel Solve Sch.Eq. for elasc wave funcon and scaerng amplude. 2

22 22 ason Scaerng Formalsm v v v v E v v we also need and, roec ou heelasc scaerng channel. Hence are equal. nuclear saes all For ansymmerc lef operaewh, nucleon he ranson amplude for arge of n erms, defne solve for, for brevy. us wre, Deals from precedng sldes mulple scaerng expanson

23 23 ason Scaerng Formalsm B C BC C BC C B C BC BC B C BC C U U k k k k k k k k we ge and hen, *Sarng wh used for he propagaor. was deny* where he operaor opcal poenal;, subsueno he above eqn., he followng for Solve : Deals from precedng sldes

24 24 ason Scaerng Formalsm see FH ppendx ~ ~ ~ ~ ~ ~ ~ where, ~ ~ ~ dd& subrac he elasc channel, 2, 2, 2,,,,, k k U U U Deals connued, herefore he non-perurbave expanson has been reorganzed n erms of wo, perurbave expansons. he frs nvolves many-body correcons o he quas-free wo-nucleon -marx; he second nvolves correcons o he opcal poenal nvolvng correlaons n he nuclear wave funcon. E v v Opcal poenal

25 KM Scaerng Formalsm Based on KM, nn. hys. 8, wh updaed noaon mulple scaerng expanson n KM 25

26 KM Scaerng Formalsm KM opcal poenal 26

27 KM Scaerng Formalsm Free-space N+N scaerng -marx 27

28 KM Scaerng Formalsm Expanson of he effecve, wo-body neracon n erms of he free-space N+N scaerng amplude 28

29 Feshbach, al and Hufner Scaerng Formalsm [coped from FH, nn. hys. NY 66, 20 97] 29

30 Feshbach, al and Hufner Scaerng Formalsm For example : v vg v vgv vgvgv v v vgv gv v gv 30

31 Feshbach, al and Hufner Scaerng Formalsm KM-marx many-body propagaor.e. average f evaluaed for an- symmercsaes Solve.6for v, sub.no.5 s redundan wh defnon of n.3 3

32 Feshbach, al and Hufner Scaerng Formalsm ypo: Leadng - order approx. okm ~ K Usng operaor deny : B B B 00 where ~ ~ because of hean-symmery mposed on N 32

33 Feshbach, al and Hufner Scaerng Formalsm KM 2 nd -order opcal poenal 33

34 Feshbach, al and Hufner Scaerng Formalsm N 00 2 nd -order poenal erm cancels Form of he 2 nd -order opcal poenal n above sldes 34

35 Feshbach, al and Hufner Scaerng Formalsm verage over parcle pars n nuclear ground-sae; ncludes correlaons roduc of sngle parcle, ground sae denses; uncorrelaed roporonal o rue, wo-parcle correlaons n he nuclear ground-sae, weghed wh he effecve proecle + nucleon neracon squared. 35

36 Brueckner effecve neracon for nuclear maer and nuclear srucure Brueckner appled he mulple scaerng heory of ason o nfne nuclear maer INM, bu wh a few changes. he goal of he INM problem s o calculae from he bare N+N neracon he bndng energy and sauraon densy of INM as esmaed by he sememprcal mass formula abou 6 MeV/ and he denses n he nerors of large nucle abou 0.7 nucleons/fm 3. he man dfference for INM from he scaerng soluon nvolves he srucure of he effecve wo-body operaors. For scaerng he dea was o formulae he many-body problem n erms of effecve wo-body operaors whch can be well approxmaed wh he free-space scaerng amplude. For INM here s no free space and we should buld he followng no he lowes-order effecve neracon operaor: aul excluson neracng pars of parcles may only ump n o, and ou of, unoccuped saes above he Ferm surface, and 2 he average nucleon poenal energy n INM n he propagaor. lso, n Brueckner s heory wo-nucleon correlaons, dscussed above, were negleced. 36

37 asympocform o be a plane wave where v For scaerng he energy denomnaor was s he N N neracon, ground saemarx elemen. Brueckner effecve neracon For INM hereare no ougong sphercal waves; he unform, soropcsymmery requres 2 2 k and heenergy o be E k U k, where m s he nucleon mass and U k s he poenalenergy. 2m lso here s no "specal" nucleon lke he proecle n scaerng. Operaors herefore refer o any arbrary par of nucleons wh parcle labels,. he wo- body operaor equaon analogous o ha for scaerng s : Behe - oldsoneequaon s he propagaor n INM,and proecs only unoccuped nermedae saes above he Ferm surface. 0 For INM heenergy denomnaor s k k E U U k v kk E H0 VC where H0 sums all knec energy operaors, 2m 2m k kk V s he poenalenergy of INM,and k, k sum over all nucleons oher han,. Inermedaesaesonly C e k k r B, v vinm B, E H H H perm excaon s of he, nucleon par; he INM remans n s ground sae,.e. we only consder he gs V C INM gs. a nucleon' s 37

38 From he algebra gs he above g.s. marx elemen of For an arbrary N N par, hs mus be solved eravely. Brueckner effecve neracon leadng hesum over all dagonal elemens where subscrp marx elemen of he opcal poenal. For INM heres no absorponand hs erm s real, and gves us he poenalenergy. he propagaorfor INM s herefore gven by V 0 f o he frs - order opcal poenalhs INM m, n boh saes 2 k 2m and and he poenalenergy n he denomnaor s normalzed he average, par - wse poenalenergy s gven by C INM gs E H B mn B mn v B, v, B, E B. E m, m, n s analogous o he frs - order, elasc - channel v mn E n 2 m E 2 2 kn 2m are above he Ferm surface, and zero oherwse mn 2 k 2m 2 m mposes an - symmercsaes. v mn per nucleon B B 2 2 kn 2m B, mn B poenalenergy s par. where, and B, 38

39 Relavsc mulple scaerng & effecve neracons Ref. J. D. Lumpe, LR hys. Rev. C 35, sem - relavsc scaerng model p m v p H E where, are he usual Drac marces for he proecle proon.he sem - relavsc propagaor of roecng he elasc channel U U v v 0 m H U 0 m E relavsc sngle - parcle saes n r Y rˆ r ~ ˆ n r Y r are he posveenergy upper and lower componens, and and followng he above sepsfor he ason opcal poenalgves, whereu, 2 and he sem - relavsc many - body - marx where E he nuclear ground sae here s consruced from an - symmerccombnaons Slaer deermnan u n D p v p p p D p gs v p mpulseapprox v D gs where Y for p n Hamlonan form : gs p v s he Drac p s an average, nermedaenuclear excaon energy, and p g D gs gs gs where gs s gven n S,,V,, form see Chp.I. rˆ s hespn- angle funcon see Chp.I, and gs ~ for opcal poenal 2. s s 39

40 hs concludes Chaper 3: Effecve Ineracon heory 40