The National Superconducting Cyclotron State University

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1 Dual Nature of Nuclei from Shell model to Isospin Diffusion Shell Model Magic numbers N=20 Liquid Drop Model Fission Process N=10 N=2 Bet t y Tsang, Tohoku Univ ersit y, Sendai, Japan The National Superconducting Cyclotron State University

2 Maria Goeppert-Mayer & Hans Jensen, 1963 Noble Prize winners for the Nuclear Shell Model. Magic number N=20 N=10 Spectroscopic Factors: measure the single structure of the valence nucleons. N=2

3 d.c.s (mb/sr) 10 S 1 0 l, Spectroscopic factors from transfer reactions j d ( ) d d ( ) d 14.3 MeV Ca40(d,p)Ca41 EX RM angle (deg) CH DWBA Hjorth Pros: We know the exact state of the nucleon transferred. Good understanding of the experimental technique and reaction theory (DWBA). Lots of data from past 40 years. Cons: Do we measure the absolute spectroscopic factors? Data appear to give inconsistent results One of the important technique to understand the structure of the rare nuclei.

4 Spectroscopic Factors from literatures Example: 1p 1/2 neutron SF in 13 C = 12 C+n Published spectroscopic factors show large fluctuations from analysis to analysis Consequence of using different optical model potentials and parameters for the DWBA reaction model.

5 S l, j d ( ) d d ( ) d EX DWBA A(d,p)B B(p,d)A Basic assumptions of DWBA The reaction is dominated by 1-step direct transfer. Elastic Scattering is the main process in the entrance and exact channels. T DWBA = < A p f V B d i >

6 Extraction of Spectroscopic Factor S l, j d ( ) d d ( ) d EX RM d.c.s (mb/sr) MeV Ca40(d,p)Ca angle (deg) CH DWBA Hjorth For each angular distribution: 1. Fit first peak only (emphasize on maximum and shape) 2. Require more than 1 data point 3. Use global proton optical potential and standardized parameters. 4. Construct d potential from p & n potential using the Adiabatic Approximation (Soper-Johnson).

7 Procedure 1. Digitize (p,d) and (d,p) angular distribution data from literature. 12 C(d,p) 13 C gs 2. Run DWBA calc s with standard parameter set. 3. Extract SF

8 Systematic extraction of SF s Liu et al, PRC 69, (2004) The spectroscopic factors deduced in a systematic and consistent way show that we can extract spectroscopic factors within the measurement uncertainties. Apply the technique to a large data set

9 We studied 79 nuclie by digitizing ~ 430 angular distributions from literature for (p,d) & (d,p) reactions on target from Z=3-24 Z=3 Li 6, 7, 8 Z=4 Be 9, 10, 11 Z=5 B 10, 11, 12 Z=6 C 12, 13, 14, 15 Z=7 N 14, 15, 16 Z=8 O 16, 17, 18, 19 Z=9 F 19, 20 Z=10 Ne 21, 22, 23 Z=11 Na 24 Z=12 Mg 24, 25, 26, 27 Z=13 Al 27, 28 Z=14 Si 28, 29, 30, 31 Z=15 P 32 Z=16 S 32, 33, 34, 35, 36, 37 Z=17 Cl 35, 36, 37, 38 Z=18 Ar 36, 37, 38, 39, 40 Z=19 K 39, 40, 41, 42 Z=20 Ca 40, 41, 42, 43, 44, 45, 47, 48, 49 Z=21 Sc 45, 46 Z=22 Ti 46, 47, 48, 49, 50, 51 Z=23 V 51 Z=24 Cr 50, 51, 52, 53, 55

10 Digitization of ~430 angular distributions from literature for (p,d) & (d,p) reactions on target from Z=3-24 Data come from many groups over 40 years. -- Require quality control How to assess the uncertainties of the procedure? S n (p,d) : S + (d,p) : S - Self Consistency Checks 79 nuclei from Li to Cr 47 nuclei 55 nuclei (p,d) & (d,p) 18 nuclei A+pB+d S + B+dA+p S - Equivalent processes S + = S -

11 Comparison of (p,d) and (d,p) reactions 10 pd vs. dp line SF(d,p) SF(p,d) By requiring the chi-square per degree of freedom is 1, we obtain nominal uncertainty of 20% for each measurement.

12 Comparison with Endt s results Endt in 1977 compiled SF s of the s-d shell nuclei from (p,d), (d,p) 50% uncertainty (p,d), (d,p), (d,t), ( 3 He, a) 25% uncertainty 10 line Endt vs. Data Data Endt's best SF There are some scattering of the values but there is a strong correlation between present analysis and Endt values

13 Textbook Example: Spectroscopic factors of Ca isotopes Direct Nuclear Reaction Theories by Austern; pg 291 l=7/2, S=1, 2, 0.75, 4, 0.5, 6, 0.25, 8 A Ca = 40 Ca +(A-40)n Assume 40 Ca is a good inert core. Ca d3/ Ca f7/ Ca f7/ Ca f7/ Ca f7/ Ca f7/ Ca f7/ Ca f7/ Ca p3/ Sc f7/ IsotopS_n s valencipm Endt shell Expt

14 spectroscopic factor IPM (Austern, pg 291) For n even S n 2 j 1 Data Austern IPM model Shell model Ca A S For n odd 1 n Ca isotopes have good single particle states with spherical cores SF for 49 Ca is lower than IPM and shell model predictions.

15 Comparison with Austern s IP Model Expt SF Most experimental SF values are less than predictions. There are no constant quenching even for close shell nuclei. Discrepancies may be explained by including interaction between nucleons and core IPM SF Be line B C N O Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr 20% line -20% line Li

16 Expt SF 0.1 Compare with Modern Shell Model (Oxbash) Good agreement with most isotopes Outliners: deformed nuclei and isotopes with small SF s 10 1 (Ne) Shell Model SF Be line B C N O Na Mg Al Si P S Cl Ar K Ca Sc Ti 20% line -20% line Li

17 Other measurements of Spectroscopic Factors (e,e p) sensitive to interior of the wave-functions Quenched by 35% compared to IP(S)M Knockout sensitive to the tail of the wave-functions Depends on Separation energy compared to SM

18 Conclusions 1. We have extracted ground state neutron spectroscopic factors for 79 (Z=3-24) nuclei Ca to 48 Ca isotopes follow the simple IPM predictions Good valence nucleons around spherical cores No quenching for gs n-orbital for the closed shell nuclei of 40Ca? 3. Most SF s fall short of IPM predictions but agree with modern day shell model calculations SM interactions take care of most of the long-range interactions. 4. There are puzzling differences of these values compared with the SF s obtained from (e,e p) and knockout reactions.

19 Problem : Disagreement between measurements MeV PRC64(2001) MeV PR164(1967) B(d,p) 12 B q cm Discrepancies larger than the quoted experimenttal errors. Shape of 1 st peak is not the same

20 Cross-comparisons weed out bad data 40 Ca(d,p) 41 Ca 10 E d =11 MeV 10 E d =12 MeV q cm Data from Lee (PR136(1964)B971) are consistently high

21 B a Aa A Z( Z 1) a C A 2/3 V S 1/ 3 ( A 2Z) A a sym 2

22 Proton Number Z B a Aa A Z( Z 1) a C A 2/3 V S 1/ 3 ( A 2Z) A a sym 2 a sym =30-42 MeV for infinite NM ( a V sym A a Neutron Number N S sym A 2 2 /3 ( A 2Z) ) 2 A Inclusion of surface terms in symmetry

23 Relevance to dilute and dense n-rich objects Stability of Neutron Star and its structure EOS?? exotics 10 km Sizes of nuclei with n-halo and n-skin 0 o 5 o

24 What is known about the EOS of symmetric matter E(, ) E(, 0) S sym () 2 Danielewicz, Lacey, Lynch (2002) Prospects are good for improving constraints further. Relevant for supernovae - what about neutron stars?

25 Experimental setup MSU, IUCF, WU collaboration Sn+Sn collisions involving 124 Sn, 112 Sn at E/A=50 MeV Miniball + Miniwall 4 multiplicity array Z identification, A<4 LASSA Si strip +CsI array Good E, position, isotope resolutions Ring Counter Annular Si+CsI array Z of projectile-like residue

26 HiRA group picture, 1999

27 Isoscaling constructed from Measured Isotopic yields T.X Liu et al. PRC 69, P T

28 Isoscaling from Relative Isotope Ratios e R 21 =Y 2 / Y 1 N / T Z n p / T MB Tsang et al. PRC 64,054615

29 Simple derivation of the isoscaling law Basic trends from Grand Canonical ensemble: Yields term with exponential dependence on the chemical potentials. Ratios to reduce sensitivity to secondary decays: Scaling parameters C, ( ) ( ) ), ( ), ( ), ( ) / exp( 1 2 ), ( / ), ( exp ), ( * int int Z N f Z N Y Z N Y T E J where Z Z N Z T Z N B Z N Z N Y HOT COLD i i i p n HOT feeding correction ( ) T Z T N C Z N Y Z N Y Z N R / / p n e ), ( ), (, T T p n /, / a

30 Isoscaling in statistical models Primary distributions show good isoscaling A 2 =186, Z 2 =75; A 1 =168, Z 1 =75 WCI statistical model working group (2004)

31 Isoscaling in Antisymmetrized Molecular Dynamical model A. Ono et al. PRC 68, (2003)

32 Isoscaling observed in many reactions Y 2 / Y 1 e ( N Z ) / T n p More Data 58 Ni+ 58 Ni 58 Fe+ 58 Fe E/A=30,40,47 Shetty et al (2003) p,4 He+ 116 Sn p,4 He+ 124 Sn E/A>1 GeV Botvina,Trautmann (2002) 86 Kr+ 116 Sn, 124 Sn 86 Kr+ 58 Ni, 64 Ni E/A=35 MeV Souliotis et al(2003) PRL, 86, 5023 (2001)

33 P b T Q Value, Sep. E E Coul E sym R 21 exp[(-s n N- S p Z)/T] P T Separation Energy E Coul E sym R 21 exp[((-s n + f n* ) N+(-S p +f p* + ) Z)/T] P T Chemical Potentials R 21 exp[(- n N- p Z)/T] E Coul E sym p n

34 Symmetry energy from AMD A. Ono et al. PRC 68, (2003) 60 Ca+ 60 Ca 48 Ca+ 48 Ca 40 Ca+ 40 Ca a depends on symmetry term interactions

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52 Isospin diffusion in the projectile-like region Basic ideas: Peripheral reactions Asymmetric collisions 124 Sn+ 112 Sn, 112 Sn+ 124 Sn -- diffusion Lijun Shi Projectile Target ( N Z) /( N Z)

53 Isospin diffusion in the projectile-like region Basic ideas: Peripheral reactions Asymmetric collisions 124 Sn+ 112 Sn, 112 Sn+ 124 Sn -- diffusion Symmetric Collisions 124 Sn+ 124 Sn, 112 Sn+ 112 Sn -- no diffusion Relative change between target and projectile is the diffusion effect Target ( N Z) /( N Z)

54 Isoscaling of mixed systems Y 21 exp(an+z)

55 Experimental: isoscaling;y 21 exp(an+z) Theoretical : = (N-Z)/(N+Z) a 1 2 (EES,SMM,AMD) Isospin Transport Ratio R i 2x x x x=experimental or theoretical isospin observable x=x R i = 1. x=x R i = -1. x x Rami et al., PRL, 84, 1120 (2000)

56 BUU predictions Lijun Shi E(, ) E(, 0)S sym () 2 S sym () Experimental results are in better agreement with predictions using hard symmetry terms

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58 Summary A lot of work has been done on isoscaling. Robust observable Seen in many different reactions Promising tool to study symmetry energy with heavy ion collisions Isospin Diffusion

59 Acknowledgements Spectroscopic factors: Jenny Lee, Xiaodong Liu Isospin Diffusion Bill Friedman P. Danielew icz, C.K. Gelbke, T.X. Liu, X.D. Liu, W.G. Lynch, L.J. Shi, R. Shomin, M.B. Tsang, W.P. Tan, M.J. Van Goethem, G. Verde, A. Wagner, H.F. Xi, H.S. Xu, Akira Ono, Bao-An Li, B. Davin, Y. Larochelle, R.T. de Souza, R.J. Charity, L.G. Sobotka, S.R. Souza, R. Donangelo

60 BUU predictions E(, ) E(, 0)S sym () 2 S sym () () Including the momentum dependence in the mean-field in BUU changes the agreement Need more experimental constraints B.-A. Li, C. B. Das, S. Das Gupta, and C. Gale Phys. Rev. C 69, (2004)

61 Isospin Diffusion from BUU

62 E sym () = Symmetry Energy e kin sym e 12.25MeV e kin sym 0 = 0.16 fm -3 ( 0) int sym 2 3 int esym 14MeV (/ 0 ) 2 14MeV (/ 0 ) 14MeV (/ 0 ) 1/3 38.5(/ 0 )21(/ 0 ) 2 int e sym / 0

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64 E(, ) E(, 0)S sym () 2 The density dependence of asymmetry term is largely unconstrained.

65 Danielewicaz, Lacey, Lynch Science, Dec 2002 Pressure and collective flow dynamics pressure contours density contours Both the elliptical and transverse flow reflect the pressure created in the collisions

66 Discrepancies in the data is larger than that quoted by the authors B(d,p) 12 B 11.8 MeV Liu at al, PRC64(2001) MeV Schiffer et al, PRev164(1967)1274

67 Discrepancies in the data is larger than the uncertainties quoted by the authors S(d,p) 34 S 12 MeV Van Der Baan, NPA173(1971) MeV Crozier et al, NPA198(1972)209

68 Problems with small SF determinations q cm 19 F(d,p) 20 F SF< MeV Fortune et al, PRC6(1972)21 Cross-sections are small and data fluctuate

69 Expt SF/Shell Compare with Shell Model (Oxbash) Good agreement with most isotopes within +20% A Be B C N O Na Mg Al Si P S Cl Ar K Ca Sc Ti Li

70 Compare with Shell Model (Oxbash) No n-separation energy dependence quenching Expt SF/Shell S_n Be B C N O Na Mg Al Si P S Cl Ar K Ca Sc Ti Li

71 Take A(d,p)A+1 stripping reaction as an example: A i can be expressed in terms of summation over the complete set of A1 : i f A i (r) f i f ( r ) A1 f is the overlap function defined as : i f r A1 A ( ) f i i f The theoretical spectroscopic factor S f is given by S i f i ( f ( r) d ) 2

72 Calculation The theoretical differential cross sections for a particular reaction were calculated by the modified version of code TWOFNR based on the DWBA model. Global Optical Model Potential and JLM Optical Model Potential were used. DWBA Theory For the reaction of A(a,b)B, the transition amplitude (T) is : For (d,p) reaction in zero-range approximation b a a i A B b f r d dr r V r T ), ( ), ( ) ( * ) ( ) ( ) ( ) ( * 0 na m l p n B B A B A A r Y r r D M J M M jm J fm MeV D ) )( )( ( 2 1 d d p d p p na jl jl A B m s m sm m s m j lsm r R S V

73 Optical-Model potential 1 d d U( r) Vc V f ( x0) 2.0VSO(.1) f ( xso) i W V f ( xw ) 4W f ( xd r dr dxd where f ( x i ) (1 e xi ) 1 and x ( r r A 3 ) / a i Thus 13 parameters are needed to be adjusted to reproduce the observed elastic scattering experiment. i 1 i ) Different sets of parameters were used for the same reaction at different energies.( Parameters are quite sensitive to the fitting procedure) Global optical model potential is used to avoid such sensitivity

74 Summary of the input parameters used in DWBA code TWOFNR (Surrey version) Source : PRC 69 (2004) DWBA Adiabatic CH JLM Proton potential Chapel-Hill [43] Chapel-Hill [43] JLM [47,48] Deuteron potential Daehnick [45] Adiabatic [53] from CH Adiabatic [53] from JLM Target r.m.s radius /density Shell model n-binding potential Woods-Saxon r 0 =1.25, a=0.65 Woods-Saxon r 0 =1.25, a=0.65 Woods-Saxon r 0 =1.25, a=0.65 Hulthen finite range factor Vertex constant D JLM potential scaling λ N/A N/A λ v =1.0 and λ w =0.8 [54] Non-Locality potentials p 0.85; n N/A; d 0.54 p 0.85; n N/A; d 0.54 p 0.85; n N/A; d 0.54 No adjustment of parameters for the entire range of isotopes

75 Digitization of ~430 angular distributions from literature for (p,d) & (d,p) reactions on target from Z=3-24 Strength lies in the numbers. To test the method in the quality control: 1. Compare to Endt s Best values when available. 2. Compare SF s derived from (p,d) and (d,p) reactions separately to estimate the uncertainties in our method.

76 Nuclear physics can be fun

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