Methods to describe direct reactions. II. Transfer reactions. Distorted wave approximations and beyond. N.K. Timofeyuk University of Surrey, UK

Transcription

1 Methods to descie diect eactions. II. Tansfe eactions. Distoted wave appoximations and eyond. N.K. Timofeyuk Univesity of Suey UK

2 Content Exact amplitude and distoted waves on appoximation Ovelap functions and thei popeties Peipheal eactions Deuteon eakup in dp eactions Remnant tem and RE effects DOM and tansfe eactions

3 Wave function of the system a x a The total wave function of the system satisfies the Schodinge equation H E 0 and contains outgoing spheical waves in all channels and incoming waves in the channel. Let us denote it The amplitude of the eaction will e detemined y the pojection of into : χ

4 Exact expession fo eaction amplitude μ H H E p j i ij The Schodinge equation can e ewitten as Multiplying fom the left y * * and integating ove ξ and ξ we get * Β ξ ξ ξ ξ ξ ξ χ μ d d E p δ χ G d e i ' ' k The solution of this equation is Geen s function

5 To get amplitude fo the a eaction the total wave function should e pojected into * Β ξ ξ ξ ξ χ d d e f e ik i ˆ k k δ χ symptotically ˆ σ k f d d v v Ω Reaction amplitude Coss section: Reaction amplitude and coss section

6 Geen s function δ χ G d e i ' ' k π μ μ ' p ' ' ik e E G 1 h To get eaction amplitude the limit should e consideed. Fo >> δ χ π μ e d ik e e i i ' k k ' h ' k ' ' ˆ ˆ ˆ π μ e f i ' k k h

7 Distoted waves Let us intoduce aitay potential U μ p U U H H E ~ ~ δ χ χ ' ' U G d ˆ χ π μ k U f h Multiplying fom the left y * * and integating ove ξ and ξ we get The eaction amplitude is then

8 1 χ μ p U G U H H E a a ˆ ˆ χ π μ π μ k k U T f h h ˆ χ χ χ k U G U U T a μ p U U H H E a The fomal solution can e witten as Then the amplitude can e ewitten as Exact amplitude contain exact wave function This wave function satisfies the Schodinge equation on appoximation

9 Distoted wave on appoximation DW T ˆ DW k χ U χ a χ is otained fom optical model in channel χ is otained fom optical model in channel x The assumption U 0 is often made. This assumption may look easonale if x is a nucleon and is lage. Then T DW k I χ k ˆ k d d χ I x a x a

10 Ovelap integals Ovelap integals cay infomation aout nuclea stuctue. They ae solutions of an integal equation. T E 0 T E 0 T E 0 T T T E E E 0 T T E E T x ε x Patial wave expansion of the ovelap integal I lm ˆ χ M M mσ 1 σ jm j jm j J M J M Ilj Ylm χ1/ σ 1/ τ

11 Popeties of the ovelap integals I symptotic ehaviou t lage the ovelap integal satisfies the equation T x εi x 0 fo neutal paticle x T x coul εi x Coul 0 fo chaged paticle x The asymptotic pat of the ovelap functions I lj is given y I lj C lj W -ηl½ κ/ C lj is the asymptotic nomalization coefficient NC W is the Whittake function κ με 1/ ε is the nucleon sepaation enegy Example: fo neuton and l0: I lj C lj exp-κ/

12 II Nomalization Definition: the nom of I lj is called the spectoscopic facto. The meaning of the spectoscopic facto fom the shell model point of view. The shell model wave function is a linea comination of the Slate deteminants D C D C { } m j l n m j l n τ τ C C γ ϕ The spectoscopic facto is expessed only via coefficients C and C which ae poaility amplitudes of a paticula shell occupation scheme.

13 Modelling the ovelap functions: T x ε x x o T x x ε I lj 0 I lj S 1/ ϕ lj 0 d ϕ lj 1 ϕ lj is the nomalized solution of the two-ody equation and the spectoscopic facto S is thought to e detemined fom expeiment. Often a standad Wood-Saxon potential with fm a 0.65 fm is used to detemine ϕ lj while the depth 0 is fitted to epoduce ε.

14 Typical example of the ovelap functions fo stale nuclei ε 5 Me l1

15 Peipheal tansfe eactions The eaction amplitude can e ewitten as follows: T DW k I χ k k d d χ ˆ 0 R cut d x... R cut d x... T in T ext T int poes the ovelap integal I lj x in the nuclea inteio. T ext poes the tail of the ovelap integal I lj x the magnitude of which is given y the NC. I lj x S 1/ ϕ lj x S 1/ lj W ηl1/ κ x / x C lj S 1/ lj lj is the single-paticle NC

16 Contiution to the dp eaction amplitude D.Y.Pang F.M.Nunes.M.Mukhamedzhanov Phys. Rev. C Cd E d 14 Me 14 Cd E d 14 Me 16 Od E d 15 Me 16 Od E d 15 Me 41 Cad E d 11 Me 41 Cad E d 11 Me Cut off in the distance etween d and fm Cut off in the distance etween n and fm The contiution of the asymptotic egion into dp eaction amplitude dominates

17 σ Coss sections of peipheal eactions ae factoized via NCs: θ ~ T int Text Text SText C Text The NC detemined fom expeiment as C ~ σ ~ exp θ / Text does not depend on. ~ ~ does not depens on The spectoscopic facto detemined fom expeiment as S σ depends on. ~ exp θ / Text Example: 1 C 8 Li 7 Li 13 C L.Tache et al Phys.Rev. C

18 Can lj e detemined fom expeimental data in a mode-independent way? Oiginal idea S..Gonchaov et al Yad.Fiz : σ θ T T ST % ST % int ext T ext can e fixed using NCs measued fom peipheal eactions. Then T int can e detemined. int does not depends on S ext T% int σ θ S T% ext T% σ θ T% S int ext σ θ σexp θ th Rth R exp S th th Cexp

19 13 Cpd 1 C E p 18.6 Me S..Gonchaov et al Yad.Fiz Rth R exp exp fm -1/ S exp R th fm a 0.49 fm fm

20 08 Pdp 09 P.M. Mukhamedzhanov and F.Nunes Phys. Rev. C fm -1/ S 0.74

21 dp eactions. eyond the on appoximation. T ˆ k χ U Exact w.f. on appoximation: χ d p d d np n d np eyond the on appoximation: taking deuteon eakup into account. np R np R T T E R 0 R np np n p np np

22 Solving 3-ody Schödinge equation in the adiaatic appoximation. Johnson-Sope model. R.C. Johnson and P.J.R. Sope Phys. Rev. C diaatic assumption: T ε R 0 np np d np np Then the thee-ody equation ecomes TR n p Ed np R np 0 Ed E εd To calculate the eaction amplitude T ˆ k χ np np R Only those pat of the wave function whee np 0 ae needed: np T R R E R0 0 R n p d np np

23 Johnson-Sope model The zeo-ange dp eaction amplitude fomally looks exactly the same as the zeo-ange DW amplitude T ZR χ R I R np x R0 The thee-ody wave function is calculated fom the two-ody Schödinge equation T R R E R0 0 R n The adiaatic inteaction potential is a sum of the poton and neuton potentials taken at half deuteon enegy p d np The model takes the deuteon eakup into account as d R0 includes all deuteon continuum states

24 Solving 3-ody Schödinge equation using Weineg state expansion. Johnson-Tandy model. R.C. Johnson and P.C. Tandy Nucl. Phys np R np i 1 ϕ i np χ i R χ i ϕ i np np Weineg asis: T np inp εd ϕi np ϕ1 ϕd ϕi np ϕ j δ ij It is assumed that only fist tem of the expansion is impotant. It can e found if all the coupling to the othe channels ae neglected T R E χ1 R 0 R d R d np ϕ 1 np np np n R np p R np

25 Johnson-Tandy model The dp eaction amplitude fomally looks exactly the same as the finite-ange DW amplitude The thee-ody wave function is calculated fom the two-ody Schödinge equation np R np The Johnson-Tandy two-ody deuteon potential is calculated in a folding pocedue involving a sum of the poton and neuton potentials taken at half deuteon enegy The model takes the deuteon eakup into account as the pojection of np np ϕ 1 np χ 1 R onto any deuteon continuum state is not zeo

26 Deuteon potential: adiaatic vs conventional J.D.Havy and R.C.Johnson Phys. Rev Conventional potential has an asoptive pat that has to account fo deuteon eakup. In adiaatic appoach this eakup is explicitly included.

27 116 Sndp 117 Sn E d 8. Me R.R. Cadmus J.and W. Haeeli Nucl. Phys Deuteon potential:

28 1 Cdp 13 C at E d 51 Me.M. Mukhamedzhanov and F. Nunes Phys. Rev. C Calculated using adiaatic d- 1 C potential DW It is not possile to detemine and theefoe S fom this gaph. diaatic coss sections ae moe peipheal than the conventional DW coss sections the contiution fom patial waves with low elative oital momentum is suppessed

29 Remnant tem in the tansfe eaction amplitude. T ˆ DW k χ U χ a The DW eaction amplitude has two tems: T DW χ ˆ k χ i j ij U x χ χ a a 1 The main tem that factoizes via SFs o NCs and depends on small x The emnant tem that does not factoize via SFs o NCs and is not ounded y small x diaatic calculation always assume that the emnant tem can e neglected.

30 ppoximate way to include emnant tem χ χ χ χ a a ij j i U U Often is chosen as a complex optical potential etween and. No theoetical justification is given to this choice. Remnant tem is impotant fo heavy paticle tansfe

31 W.R. Smith Nucl. Phys Compaing emnant and no-emnant calculations 1 Cdp 13 C E d 8 Me np p - p np Compaing vaious veions of emnant calculations 8 Sidp 9 Si np p - p Δ 0np -4 Me W p0 E d 10 Me

32 voiding calculation of the emnant tem a x x U T χ χ ˆ k U is aitay. Let us choose instead of U μ H H E p j i ij μ p U U H H E ij j i Reminde: deivation of eaction amplitude: μ ij j i ij j i H H E p μ μ x x x x x H H H E p p x φ x T ~ ˆ k 0 ~ φ μ μ x x x x H H H E p p x x

33 Recoil excitation and eakup 0 x 0 x x x x 0 T ˆ k ~ φ x ~ T E φ 0 x What does it mean? ~ eakup of is included in φ as its ovelap with any continuum state of is not zeo. The eakup of occus ecause inside inteacts with gets a ecoil and then passes it via x to x. The pice to pay fo getting id of the emnant tem is to include ecoil excitation and eakup of.

34 ~ Wave function φ in the adiaatic model: ~ φ χ k x x iμ k x e μ m / m x x m In the zeo-ange appoximation x x x x D0δ x the a eaction amplitude ecomes T ZR ad iμ k * D 0 d χ k e x χ k In the conventional appoach that assume that the emnant tem is small T ZR standad * ~ D 0 d χ k μ x χ k ~ μ m / m m x

35 N.K. Timofeyuk and R.C.Johnson Phys. Rev. C

36 Tansfe eactions with dispesive optical potentials N.. Nguyen et al Phys. Rev. C ' E 0 ' Δ ' E iw ' E opt Δ ' E P π de' ' E' W E E' DOM fom has een descied in tems of 3 paametes used to fit data sets fo and 4 E 00 Me taken fom J.M. Muelle et al Phys. Rev. C SnN 13 SnN

37 13 Sndp 133 Sn E d 9.46 Me Johnson-Tandy adiaatic model has een used to calculate tansfe coss sections emnant tem is neglected. DOM can also pedict potential well fo neuton ound state Standad WS DOM 13 Snn

38 NCs otained fom tansfe eactions using Gloal systematic of nucleon optical potentials CH89 DOM Woods-Saxon potential used fo neuton ound state DOM used fo neuton ound state

39 Spectoscopic factos otained using Gloal systematic of nucleon optical potentials CH89 DOM Woods-Saxon potential used fo neuton ound state DOM used fo neuton ound state

40 Summay Exact amplitude can e witten using distoted waves T ˆ k χ x U on appoximation fo exact wave function can e intoduced that selects only one channel of inteest Tansfe eactions poe ovelap functions. Many tansfe eactions ae peipheal they ae sensitive only to the asymptotic pat of the ovelap integal given y NCs. If potential well fo ound state ae elialy detemined then spectoscopic factos can e studied in tansfe eactions as well. In the dp eaction deuteon eakup channels play impotant ole. They can appoximately e taken into account using Johnson-Sope and Johnson- Tandy appoximations. These appoximations allow availale DW codes to e used. Remnant tem can e exactly excluded fom tansfe eaction calculations ut then ecoil excitation and eakup effects in the final nucleus should e taken into account Tansfe eactions enefit fom using optical potentials otained fom DOM. Then NCs and SFs ae less dependent on incident enegies.