Experimental Evidence for Magic Numbers Ying CHEN ( L) SupervisorµProf. Jie MENG (Š#) School of Physics, Peking University June 7, 211 CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Outline Outline 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
Outline Introduction 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Introduction Introduction The nuclei exhibit extra stability when either proton number Z or neutron number N has the value 2, 8, 2, 28, 5, 82, or 12. These numbers are the so-called magic numbers. The difference between neutron separation energy observed and calculated with semi-empirical mass formula in liquid drop model (LDM): S n = S n (A, Z) exp S n (A, Z) calc Sharp discontinuities are evident for N = 5, 82 and 12 showing a sudden decrease in separation energy of (N + 1)th neutron, which reflects the existence of well-known magic number N R. D. Evans, The Atomic Nucleus, 1955. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Introduction Introduction The odd-even effect brings in discontinuities to S n which are not associated to shell closure. (S 2n ) EXP (S 2n ) LDM is more appropriate for shell closure study as two related nuclei in S 2n have the same neutron number parity. (S n ) E X P - (S n ) L D M [M e V ] 2-2 - 2 8 1 1 2 1 1 N e u tr o n N u m b e r N u c le i w ith Z = e v e n This work Appearance of new magic numbers and disappearance of known magic numbers are studied through (S 2n ) EXP (S 2n ) LDM with new experimental data (G. Audi et al., NPA 729 (2) 7). CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 5 / 55
Outline Neutron separation energy and magic number 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Neutron separation energy and magic number General formulas Difference between two-neutron separation energy observed and calculated with semi-empirical mass formula in LDM: S 2n(2p) = (S 2n(2p) ) EXP (S 2n(2p) ) LDM. (1) a. Two-nucleon separation energy S 2n(2p) is defined as From the semi-empirical mass formula in LDM: S 2n (A, Z) = BE(A, Z) BE(A 2, Z) (2) S 2p (A, Z) = BE(A, Z) BE(A 2, Z 2). () BE(A, Z) = a V A a S A 2/ a C Z(Z 1)A 1/ a A (A 2Z) 2 A 1, () S 2n (A, Z) can be written as S 2n (A, Z) 2(a V a A ) a SA 1/ + 2 a CZ(Z 1)A / + 8a A Z 2 A(A 2), where the values for the LDM parameters: a V = 15.85 MeV, a C =.71 MeV, a S = 18. MeV and a A = 2.21 MeV. A. H. Wapstra et al., Nucl. Data Tables 9 (1971) 27. b. The experimental data of two-neutron separation energy is taken from G. Audi et al., Nucl. Phys. A 729 (2) 7. A magic number appears as a sudden decrease in (S 2n ) EXP (S 2n ) LDM at the neutron number (N + 2) following the neutron magic number N. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 7 / 55
Neutron separation energy and magic number Neutron separation energy and magic number (S 2 n ) E X P -(S 2 n ) L D M [M e V ] (S 2 p ) E X P -(S 2 p ) L D M [M e V ] 9 - - -9 - - N e u tr o n N u m b e r 2 8 1 1 2 1 1 8 e v e n -e v e n n u c le i 2 8-9 5 2 8 2 1 2 2 8 1 1 2 1 1 P r o to n N u m b e r (S 2 n ) E X P -(S 2 n ) L D M [M e V ] (S 2 p ) E X P -(S 2 p ) L D M [M e V ] 9 - - 9 - - -9 N e u tr o n N u m b e r 2 8 1 1 2 1 1 o d d -e v e n n u c le i 2 8 5 8 2 8 2 1 2 2 8 1 1 2 1 1 P r o to n N u m b e r Sudden decreases of the order of MeV are evident for N = 28, 5, 82, 12, which reflects the existence of known magic number. Two-neutron separation energy differences are complicated in light nuclei, thus further study in this region is needed. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 8 / 55
Outline Light nuclei 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 9 / 55
Odd-even nuclei Light nuclei (S 2 n ) E X P -(S 2 n ) L D M [M e V ] F N a A l P C l K S c V 1 2 N e u tr o n N u m b e r Figure: (S 2n ) EXP (S 2n ) LDM for odd-even nuclei as a function of neutron number. Well-known magic number a. N = 8: F isotope chain, no decrease, indicating disappearance of magic number; b. N = 2: Na and Al isotope chains, no decreases; P to Sc isotope chains, sudden decreases of.55,.51, 2.7 and 2.19 MeV respectively; From Na to Sc isotopes, shell closure appears for Z 15 and shell effect first increases and then decreases with Z; c. N = 28: K to V isotope chains, sudden decreases of 1.92, 2.8 and 1.5 MeV respectively. Shell effect first increases and then decreases with Z; Possible new magic number a. N = 1: Al and P isotope chains; b. N = 1: F isotope chain; c. N = 2: P isotope chain; Sudden decreases exist, an indication of new magic number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Even-even nuclei Light nuclei (S 2 n ) E X P -(S 2 n ) L D M [M e V ] - O N e M g S i S A r C a T i 1 2 N e u tr o n N u m b e r Figure: (S 2n ) EXP (S 2n ) LDM for even-even nuclei as a function of neutron number. Well-known magic number a. N = 8: O isotope chain, sudden decrease of 5.19 MeV ; Ne isotope chain, no decrease; b. N = 2: Mg, Si, Ar and Ti isotope chains, no decreases; S and Ca isotope chains, sudden decreases of.51 and.7 MeV respectively; From Mg to Ti isotopes, similar evolution of shell effects as odd-even nuclei is absent; c. N = 28: Ca and Ti isotope chains, sudden decreases of.7 and 2.5 MeV respectively. Possible new magic numbers a. N = Z: Sudden decrease is evident in each isotope chain, which is caused by special properties of nuclei with N = Z or shell effect; b. N Z region N = 1 Ne and Mg isotope chains; N = 2, S isotope chain; N = 2 Ti isotope chain; Sudden decreases exist, an indication of new magic number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 11 / 55
Outline Light nuclei Even-even nuclei with N = Z 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 12 / 55
Extra binding energy Light nuclei Even-even nuclei with N = Z Surfaces of experimental masses of even-even and odd-odd nuclei exhibit a sharp discontinuity at N = Z. This cusp, reflecting an additional binding in nuclei with neutrons and protons occupying the same shell model orbitals, is knownly attributed to neutron-proton pairing correlations. W. Satula et al., PLB 7 (1997) 1, E. P. Wigner, Phys. Rev. 51 (197) 1. In order to understand the reason of sudden decrease of (S 2n ) EXP (S 2n ) LDM at N = Z, the extra binding should be removed first. Assumptions of extra binding energy extraction Chasman s method R. R. Chasman, PRL 99 (27) 8251 : a. Extra binding energy is related to N Z ; Present method in even-even nuclei: a. Extra binding energy occurs in N = Z nuclei; The binding energies in these nuclei are decomposed into a part B which varies smoothly with N and Z and extra binding energy. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Extract extra binding energy Light nuclei Even-even nuclei with N = Z Double-difference formula δv, involving smooth contribution δ S 2n (Z, N) and extra binding, is written as δv (Z, N) = [B(Z, N) B(Z, N 2)] [B(Z 2, N) B(Z 2, N 2)] = δ S 2n (Z, N) + 2, (5) where δ S 2n (Z, N) = [ B(Z, N) B(Z, N 2) ] [ B(Z 2, N) B(Z 2, N 2) ]. Average double-difference formula, δv, is defined as δv (Z, N) = 1 [δv (Z 2, N) + δv (Z + 2, N) + δv (Z, N 2) + δv (Z, N + 2)] = δ S 2n (Z, N), where δ S 2n (Z, N) = 1 [ δ S2n (Z 2, N) + δ S 2n (Z + 2, N) + δ S 2n (Z, N 2) + δ S 2n (Z, N + 2) ]. () If we consider δ S 2n (Z, N) approximately equals to δ S 2n (Z, N), from Eq. (5) and Eq. () extra binding energy is extracted as = 1 [ ] δv (Z, N) δv (Z, N). (7) [ ] Result of Chasman s method: = 1 δv (Z, N) δv (Z, N). CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Extra binding energy Light nuclei Even-even nuclei with N = Z E v e n -e v e n n u c le i w ith N = Z P re s e n t m e th o d C h a s m a n s m e th o d D (Z,N ) [M e V ] 2 1 1 2 N e u tr o n N u m b e r Figure: Extra binding energy in N = Z even-even nuclei as a function of neutron number. The black line and red line are obtained from the present method and Chasman s method respectively. The extra binding energies extracted with both methods have the same behavior with increasing neutron number; The gap between the results of two methods is attributed to different assumptions. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 15 / 55
Light nuclei Even-even nuclei with N = Z Neutron separation energy after removing extra binding (S 2 n ) E X P -(S 2 n ) L D M [M e V ] - O In c lu d in g D (Z,N ) N e M g S i S A r (S 2 n ) E X P -(S 2 n ) L D M [M e V ] - - O s u b tr a c tin g D (Z,N ) N e M g S i S A r C a C a T i T i 1 2 N e u tr o n N u m b e r 1 2 N e u tr o n N u m b e r N = Z for non-magic nuclei: no decreases remain, no indication of new magic numbers; N = 2 for Ar and Ti isotope chains: Recovering of decreases indicates the existence of this magic number. From Mg to Ti isotopes, shell closure appears for Z 1; shell effect first increases and then decreases with Z and reaches a maximum at double magic nucleus Ca; N = 1 for S isotope chain and N = 1 for Si isotope chain: Occurrence of new decreases indicates appearance of new magic numbers; N = Z for magic nuclei: Decrease remains for Ca, but disappears for 1 O, because the extra binding energy extracted with present method partly mixes the shell effect in very light nuclei. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Outline Light nuclei Possible new magic number and disappearance of known magic number 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 17 / 55
Light nuclei Possible new magic number and disappearance of known magic number Possible new magic number and disappearance of known magic number P r o to n N u m b e r 2 1 O M g N e S i S A r C a T i F N = 2 N a V S c K C l P A l N = 2 8 s ta b le n u c le i p o s s ib le n e w m a g ic n u c le i d is a p p e a ra n c e o f u s u a l m a g ic n u c le i N = 8 u n d e fin e d re g io n 1 2 N e u tr o n N u m b e r Figure: A partial nuclear chart indicating the disappearance of known magic number and appearance of new magic number. Disappearance of known magic number: a. N = 8, F and Ne isotopes, b. N = 2, Na to Si isotopes. Possible new magic number: a. N = 1, Al, P and S isotopes, b. N = 1, F, Ne, Mg and Si isotopes, c. N = 2, P and S isotopes, d. N = 2, Ti isotope. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 18 / 55
Outline Summary 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 19 / 55
Summary Summary Difference between two-neutron separation energy observed and calculated with semi-empirical mass formula in LDM is calculated for all even-even and odd-even nuclei in Audi 2 mass table. Well-known magic numbers N = 5, 82, 12 are observed in medium heavy nuclei. Appearance of new magic numbers and disappearance of known magic numbers in light nuclei are studied: a. Indicate the disappearance of known magic numbers: N = 8, F and Ne isotopes; N = 2, Na to Si, Ar and Ti isotopes; possible new magic numbers: N = Z, all even-even nuclei; N = 1, Al and P isotopes; N = 1, F, Ne and Mg isotopes; N = 2, S isotope; N = 2, Ti isotope; b. A method to extract extra binding for even-even nuclei with N = Z is proposed; c. Indicate disappearance of known magic numbers N = 8, 2 after removing extra binding: N = 8, F and Ne isotopes; N = 2, Na and Si isotopes; possible new magic numbers N = 1, 1, 2, 2: N = 1 Al, P and S isotopes; N = 1 F, Ne, Mg and Si isotopes; N = 2 P and S isotopes; N = 2 Ti isotope. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
Summary CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 21 / 55
Outline Appendix 1 Introduction 2 Neutron separation energy and magic number Light nuclei Even-even nuclei with N = Z Possible new magic number and disappearance of known magic number Summary 5 Appendix CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 22 / 55
o( Appendix é Audi 2 ŸþL kóóø ÛóØ OŽV f lu Š %.Œ²ŸþúªOŽŠƒ Ÿþ«ïÄDÚŽêž #Žê)µ a. * DÚŽê N = 8, 2 ž µ N = 8 F Ne Ó ƒó N = 2 Na Si, Ar Ti Ó ƒó ±9ŒU#Žê)µ N = Z k N = Z óóø N = 1 Al P Ó ƒ N = 1 F, Ne Mg Ó ƒ N = 2 S Ó ƒ b. é N = Z óóø ÑJ åpu { c. žø åpu ý«dúžê N = 8, 2 ž µ N = 8 F Ne Ó ƒ N = 2 Na Si Ó ƒ ±9ŒU#Žê N = 1, 1, 2, 2 )µ N = 1 Al, P 9 S Ó ƒ N = 1 F, Ne, Mg Ú Si Ó ƒ N = 2 P S Ó ƒ N = 2 Ti Ó ƒ CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
Appendix V f luƒ Ÿfêüz D S 2 n (Z,2 )-D S 2 n (Z,2 2 ) [M e V ] D S 2 n [M e V ] 2 1-1 2 N = 2 N = 2 2 D S 2 n (Z,2 8 )-D S 2 n (Z, ) [M e V ] D S 2 n [M e V ] 2 1 2 N = 2 8 N = -2-2 1 2 1 1 1 8 2 2 2 P r o to n N u m b e r 1 8 2 2 2 2 2 2 8 P r o to n N u m b e r CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
ŒU#Žê Appendix (S 2 n ) e x p -(S 2 n ) L D M [M e V ] 2-2 R b S r Y Z r N b 8 5 2 5 N e u tr o n N u m b e r N = 5? é Sr Nb Ó ƒó V f luƒññy²wac DÚŽê N = 5 é Sr Nb Ó ƒó, á N = 5? é Sr Nb Ó ƒó V f luƒññy²wac Œ.7 MeV,.58 Mev,.1 Mev,.98 MeV,.72 MeV N = 5 é Sr Nb Ó ƒó ŒU #Žê CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 25 / 55
ŒU#Žê Appendix.5 (S 2 n ) e x p -(S 2 n ) L D M [M e V ]. -.5-1. -1.5 C m C f F m N o B k E s -2. 1 1 1 8 1 5 2 1 5 1 N e u tr o n N u m b e r N = 152? é Cm No Ó ƒó V f luƒññy²wa C é Cm No Ó ƒ N = 152 ŒU #Žê CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
åpuj Appendix Surface of experimental masses of even-even and odd-odd nuclei exhibit a sharp slope discontinuity at N = Z. This cusp, reflecting an additional binding in nuclei with neutrons and protons occupying the same shell model orbitals, is usually attributed to neutron-proton pairing correlations. þ ÑóóØ ÛÛØŸþ N = Z?LyѲwÇØëY5 ù ØëY:Ï~@ d f-ÿfé'éúå NŸf fóâƒó Š.;fØ åpu E. P. Wigner, phys. Rev. 51 (197) 1, W. Satula, PLB 7 (1997) 1. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 27 / 55
Appendix V f lulˆªí føvøf luœ²ÿþúªµ l %. LDM Œ²;Ÿþúªµ Œ±µ S 2n (A, Z) = BE(A, Z) BE(A 2, Z) (8) S 2p (A, Z) = BE(A, Z) BE(A 2, Z 2). (9) BE(A, Z) = a V A a S A 2/ a C Z(Z 1)A 1/ a A (A 2Z) 2 A 1, (1) S 2n (A, Z) 2(a V a A ) a SA 1/ + 2 a CZ(Z 1)A / + 8a A Z 2 A(A 2), y²l Xeµ NÈ µ a V A a V (A 2) = 2a V, (11) é U µ a A (A 2Z) 2 A + a A (A 2Z 2) 2 A 2 = 2a A + 8a A Z 2 A(A 2). (12) ±þü zož î " CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 28 / 55
Appendix V f lulˆªí L zµ a S A 2/ a S (A 2) 2/ ( A) d(a SA 2/ ) da (1) = a SA 1/, (1) Õ µ a c Z(Z 1)A 1/ + a C Z(Z 1) = a c Z(Z 1)( 2 A / ). (15) L Õ íæ^cqµòl 9 Õ 9ŸþêCz È5OŽ" zcq^ùê CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 29 / 55
Chasman óš Appendix E x tr a B in d in g [M e V ] C h a s m a n s M e th o d -2-2 -2-2 N -Z - -2 2 IPµZ = N fø Øf z²þš IP " åpu µ ²wÜ µ δv (Z, N) = [B(Z, N) B(Z, N 2)] [B(Z 2, N) B(Z 2, N 2)] = δ S 2n (Z, N) +, (1) δv (Z, N) = 1 [δv (Z 2, N) + δv (Z + 2, N) + δv (Z, N 2) + δv (Z, N + 2)] åpuµ = δ S 2n (Z, N), = 1 (17) [ δv (Z, N) δv (Z, N) ]. (18) CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
óš Appendix P r e s e n t W o r k E x tr a B in d in g [M e V ] D D N -Z - -2 2 IPµZ = N fø Ÿfê fêƒú\ åpu IP " åpu µ ²þŠµ δv (Z, N) = [B(Z, N) B(Z, N 2)] [B(Z 2, N) B(Z 2, N 2)] = δ S 2n (Z, N) + 2, (19) δv (Z, N) = 1 [δv (Z 2, N) + δv (Z + 2, N) + δv (Z, N 2) + δv (Z, N + 2)] Ñ åpuµ = δ S 2n (Z, N), = 1 (2) [ δv (Z, N) δv (Z, N) ]. (21) CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Appendix f luƒ V f luƒ O O (S n ) E X P -(S n ) L D M [M e V ] N e M g S i S (S 2 n ) e x p -(S 2 n ) L D M [M e V ] N e M g S i S A r A r C a C a 1 2 N e u tr o n N u m b e r 1 2 N e u tr o n N u m b e r Figure: The values of (S n ) EXP (S n ) LDM and (S 2n ) EXP (S 2n ) LDM as a function of neutron number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
Appendix Subtract extra binding energy for N = Z even-even nuclei (S 2 n ) E X P -(S 2 n ) L D M [M e V ] C r F e N i Z n G e S e K r S r (S 2 n ) E X P -(S 2 n ) E x tr a -(S 2 n ) L D M [M e V ] C r F e N i Z n G e S e B r S r 2 5 N e u tr o n N u m b e r 2 5 N e u tr o n N u m b e r Figure: Experimental values of (S 2n ) EXP (S 2n ) LDM and (S 2n ) EXP (S 2n ) Extra (S 2n ) LDM for Cr to Sr isotopes as a function of neutron number. N = Z šžê? acž N = Z Žê? ac 5 Ni?, N = Z + 2? #ac Fe, Ni, Zn, Se 9 Sr Ó ƒó) N = Z šdúžê? acø #Žê&Ò d åpu ÏLù {J åpu ¹Ù A" CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
ÛóØ Appendix (S 2 n ) E X P -(S 2 n ) L D M [M e V ] F N a A l P C l K S c V 1 2 N e u tr o n N u m b e r Figure: The values of (S 2n ) EXP (S 2n ) LDM for odd-even nuclei as a function of neutron number. DÚŽêµ N = 28? K V Ó ƒóñy²wa C ÙŒ g µ1.92mev, 2.8MeV, 1.5MeV; N = 2? P Sc Ó ƒóñyac ÙŒ g µ.55mev,.51mev, 2.7MeV, 2.19MeV é Na Ú Al Ó ƒó %vkacñy ŒU Nd DÚŽêž N = 8? F Ó ƒóvkñyac Ó ŒU NŽêž ŒU#Žêµ N = 1? Al P Ó ƒóñyac Œ g µ.9mev, 1.9MeV N = 1? éf Ó ƒóñyac Œ 2.MeV N = 2? P Ó 1.8MeV ƒóñyac Œ d?acœu X#Žê)" CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Appendix New magic number in heavy nuclei (S 2 n ) e x p -(S 2 n ) L D M [M e V ] 2-2 e v e n -e v e n n u c le i 1 2 The theoretical fission thresholds reveal the magicity of the neutron number 152. H. C. Pauli and T. Ledergerber, NPA 175 (1971) 55. Deformed shell closures are found at N = 152 through a systemic study of global properties of superheavy nuclei in the framework of macroscopic-microscopic method. A. Baran and K. Sieja et al, PRC 72 (25) 1. (S 2 n ) e x p -(S 2 n ) L D M [M e V ] 2-2 - 1 5 2 o d d -A n u c le i 1 1 2 1 1 (S 2 n ) e x p -(S 2 n ) L D M [M e V ].5. -.5-1. -1.5 C m C f F m N o B k E s N e u tr o n N u m b e r -2. 1 1 1 8 1 5 2 1 5 1 N e u tr o n N u m b e r CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 5 / 55
References Appendix The halo structure of a light nucleus is shown to be associated with the closed shell effects of its core nucleus, and new magic numbers at N = and 1 or 1 and/or Z = and 1 are predicted on the basis of the potential energy surfaces calculated within the cluster-core model. Raj K Gupta, M Balasubramaniam et al, J. Phys. G: Nucl. Part. Phys. 2 (2) 55. Theoretical, experimental and estimated single-neutron separation energies and two-neutron separation energies of Mg isotopes. Qijun Zhi, Zhongzhou Ren, PLB 8 (2) 1. For nuclei having approximately equal numbers of neutrons and protons, there are large changes in binding energy due to many-body effects-even for closed shell nuclei. There are also large correlation energy effects in the even-odd and odd-even nuclei, with one neutron or one proton less than the N = Z even-even nucleus. A previously derived result, that the diagonal correlation energy in the last occupied orbital for the latter nuclei is 1/2 of the equivalent correlation energy for closed shell nuclei, is tested against observed binding energies and found to be fairly accurate.r. R. Chasman PRL 99 (27) 8251. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Appendix Jumps for even-even and odd-even nuclei Table: Jumps (in MeV) in each isotope chain N 8 1 12 1 1 18 2 2 28 C.2 N 2. O 5.19. F 2. Ne 2.19. Na.2 Mg..1 Al.9 Si.8 P 1.9.55 1.8 Cl..51 S 2.28.51.8 Ar 2.25 K 2.7 1.92 Ca.7.7 CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 7 / 55
Appendix Jumps for even-even and odd-even nuclei N 2 22 2 2 28 2 8 5 5 5 Sc 2.19 2.8 Ti 1. 2.5.5 V.9 1.5 Cr 2..1 Mn.78.8.1 Fe 2. Co 2.8 Ni 5.5 Cu.25.8 Zn 1.2 1.8 Ga.5.8 Ge 1.92 As.5.2 Se 1.91 2.2 CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 8 / 55
Appendix Jumps for even-even and odd-even nuclei N 2 22 2 2 28 2 8 2 5 5 5 Br 1. 2.87.59 Kr 1.27 2.89 Rb 1.1.7.7 Sr 2.15.79.58 Y 1.8..1 Zr 2.9.81.98 Nb 1.8.7.72 There is no experimental data about nuclei with N = Z when Z >. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 9 / 55
Appendix Jumps (in MeV) for even-even and odd-even nuclei N 5 82 N 82 92 9 9 98 1 12 1 Mo. La.51 Tc.8.25 Ce.1 Ru 2.9 Pr.2 Rh.1 Nd 2.95.1 Pd.1 Pm.17.21 Ag 2.9 Sm 2.97. Cd 2.9 Eu.21.28 Sn 5. Gd.15.11.15 Sb 5. Tb.11 Te. Dy 2.82.15 I. Ho 2.87.8 Xe.7 Er 2.7.15.2 Cs.92 Tm 2.5.17.29 Ba. Yb 2.55.1. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Appendix Jumps for even-even and odd-even nuclei N 1 18 11 12 12 N 12 12 18 1 12 1 1 15 152 Lu.5 Ra.2 2.29 Hf.9.17 Ac.27 2.22.17 Ta.5 Th. 1.8 W.5 Pa.5 1..27 Re.59 U.25 Os.19 Np.2.22 Hg.51 Pu.19.19.1 Ti.2. Am.21.2 Pb.12.1 Cm.22. Bi.19.2 Bk.2.55 Po.2.8 Cf.22.. At.2.2 Es.9 Rn.27 2.8 Fm.8 Fr.8 2. No.7 CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 1 / 55
Appendix Two-neutron separation energy difference in light mass region 1. Even-even nuclei: 9 (S 2n ) exp -(S 2n ) LDM [MeV] - - C O Ne Ar Mg Si S even-even nuclei Ca Ti Cr -9 1 2 5 Neutron Number CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 2 / 55
Appendix Changes of magic number in light neutron-rich nuclei Proton Number 2 1 C O Ne Mg classic shells Ti Ca Ar S Si N F Cr Al Na P Cl K Sc stable nuclei V new magic number disappearance of usual magic number 1 2 Neutron Number CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Appendix Contribution of Wigner term In order to take account of the excessive binding of nuclei for which N Z, a phenomenological Wigner term of the form E W = V W e ( λ N Z /A) (22) was also include, where the parameters read: V w = 2.9 MeV, λ = 28.. M. Samyn and S. Goriely et al, NPA 7 (22) 12. The binding energy including Wigner term reads: BE = BE LDM + BE Wig. (2) The two-neutron separation energy including Wigner term reads: S 2n = (S 2n ) LDM + (S 2n ) Wig. (2) CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Appendix Gap for nuclei in heavy mass region.5 e v e n -e v e n n u c le i The theoretical fission thresholds reveal the magicity of the neutron number 152. (S 2 n ) e x p -(S 2 n ) L D M [M e V ]. -.5-1. H. C. Pauli and T. Ledergerber, NPA 175 (1971) 55. Gap on approach of N=152: nuclei proton number gap (MeV) -1.5 28 Cm 9.5517.5 o d d -A n u c le i 1 5 2 29 Bk 97.581 (S 2 n ) e x p -(S 2 n ) L D M [M e V ]. -.5-1. 25 Cf 98.57 251 Es 99.8517 252 Fm 1.8581 25 No 12.72-1.5 1 1 5 1 N e u tr o n N u m b e r CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 5 / 55
Appendix Gap for nuclei in heavy mass region The theoretical fission thresholds reveal the magicity of the neutron number 152. H. C. Pauli and T. Ledergerber, NPA 175 (1971) 55..5 Gap on approach of N=152: (S 2 n ) e x p -(S 2 n ) L D M [M e V ]. -.5-1. -1.5 C m C f F m N o B k E s nuclei proton number gap (MeV) 28 Cm 9.5517 29 Bk 97.581 25 Cf 98.57-2. 1 1 1 8 1 5 2 1 5 1 N e u tr o n N u m b e r 251 Es 99.8517 252 Fm 1.8581 25 No 12.72 CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 / 55
Appendix (S 2n ) EXP (S 2n ) LDM for all nuclei 9 (S 2n ) exp -(S 2n ) LDM [MeV] - - -9 2 8 1 12 1 1 Neutron Number Figure: The differences between two-neutron separation energies obtained from experiment and liquid drop model calculation for all nuclei as a function of neutron number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 7 / 55
Appendix (S 2n ) EXP (S 2n ) LDM for odd-a nuclei 9 (S 2n ) exp -(S 2n ) LDM [MeV] - - -9 2 8 1 12 1 1 Neutron Number Figure: The differences between two-neutron separation energies obtained from experiment and liquid drop model calculation for odd-a nuclei as a function of neutron number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 8 / 55
Appendix (S 2n ) EXP (S 2n ) LDM for odd-odd nuclei 9 (S 2n ) exp -(S 2n ) LDM [MeV] - - -9 2 8 1 12 1 1 Neutron Number Figure: The differences between two-neutron separation energies obtained from experiment and liquid drop model calculation for odd-odd nuclei as a function of neutron number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 9 / 55
Appendix (S 2n ) EXP (S 2n ) LDM for even-even nuclei 9 (S 2n ) exp -(S 2n ) LDM [MeV] - - -9 2 8 1 12 1 1 Neutron Number Figure: The differences between two-neutron separation energies obtained from experiment and liquid drop model calculation for even-even nuclei as a function of neutron number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 5 / 55
Appendix Experimental Evidence for Shell Effects in N=*2 Region 9 (S 2n ) exp -(S 2n ) LDM [MeV] - - -9 C O Ne Mg Si S Ar Ca Ti Ge Fe Ni 8 12 1 2 2 28 Neutron Number Figure: The differences between two-neutron separation energies obtained from experiment and liquid drop model calculation for even-even nuclei as a function of neutron number in N=*2 region. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 51 / 55
Appendix Experimental Evidence for Shell Effects in N=*2 Region 12 1 (S 2n ) exp -(S 2n ) LDM [MeV] - - O 8 Mg Si Ca 2 Fe 28 8 12 1 2 2 28 2 Neutron Number Figure: The differences between two-neutron separation energies obtained from experiment and liquid drop model calculation for even-even nuclei as a function of neutron number in N=*2 region. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 52 / 55
Appendix (S 2p ) EXP (S 2p ) LDM for all nuclei 9 (S 2p ) exp -(S 2p ) LDM [MeV] - - -9 2 8 1 Proton Number Figure: The differences between two-proton separation energies obtained from experiment and liquid drop model calculation for all nuclei as a function of proton number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 5 / 55
Appendix (S 2p ) EXP (S 2p ) LDM for even-even nuclei (S 2p ) exp -(S 2p ) LDM [MeV] - - 2 8 1 Proton Number Figure: The differences between two-proton separation energies obtained from experiment and liquid drop model calculation for even-even nuclei as a function of proton number. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 5 / 55
Appendix Experimental Evidence for Shell Effects in Z=*2 Region 28 Si 52 Fe 2 Mg (S 2p ) exp -(S 2p ) LDM [MeV] 2-2 - 1 C 8 2 22 Mg Ne Si S Ar 2 28-8 12 1 2 2 28 2 Proton Number Figure: The differences between two-proton separation energies obtained from experiment and liquid drop model calculation for even-even nuclei as a function of proton number in Z=*2 region. CHEN Ying () Experimental Evidence for Magic Numbers June 7, 211 55 / 55