Symmetry Energy in Surface

Transcription

1 Symmetry Energy in Nuclear Surface Natl Superconducting Cyclotron Lab, Michigan State U Workshop on Nuclear Symmetry Energy at Medium Energies Catania & Militello V.C., May 28-29, 2008

2 Charge Symmetry & Charge Invariance Charge symmetry: invariance of nuclear interactions under n p interchange n isoscalar quantity F does not change under n p interchange. Example: nuclear energy. Expansion in η = (N Z )/ for smooth F, has even terms only: F(η) = F 0 + F 2 η 2 + F 4 η n isovector quantity G changes sign. Example: ρ np (r) = ρ n (r) ρ p (r). Expansion with odd terms only: G(η) = G 1 η + G 3 η Note: G/η = G 1 + G 3 η Charge invariance: invariance of nuclear interactions under rotations in n-p space

3 Charge Symmetry & Charge Invariance Charge symmetry: invariance of nuclear interactions under n p interchange n isoscalar quantity F does not change under n p interchange. Example: nuclear energy. Expansion in η = (N Z )/ for smooth F, has even terms only: F(η) = F 0 + F 2 η 2 + F 4 η n isovector quantity G changes sign. Example: ρ np (r) = ρ n (r) ρ p (r). Expansion with odd terms only: G(η) = G 1 η + G 3 η Note: G/η = G 1 + G 3 η Charge invariance: invariance of nuclear interactions under rotations in n-p space

4 Charge Symmetry & Charge Invariance Charge symmetry: invariance of nuclear interactions under n p interchange n isoscalar quantity F does not change under n p interchange. Example: nuclear energy. Expansion in η = (N Z )/ for smooth F, has even terms only: F(η) = F 0 + F 2 η 2 + F 4 η n isovector quantity G changes sign. Example: ρ np (r) = ρ n (r) ρ p (r). Expansion with odd terms only: G(η) = G 1 η + G 3 η Note: G/η = G 1 + G 3 η Charge invariance: invariance of nuclear interactions under rotations in n-p space

5 Symmetry Energy: From Finite to System Skyrme Interactions η = (ρ n ρ p )/ρ expansion under n p symmetry ( ρn ρ p E(ρ n, ρ p ) = E 0 (ρ)+s(ρ) ρ S(ρ) = aa V + L ρ ρ ρ 0 Finite Nucleus Nucleon densities ρ p (r) & ρ n (r) Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 +a (N Z )2 a() +E mic a a? = a V a a a = a V a + 2/3 a S a a a () =? half-infinite matter ) 2

6 Symmetry Energy: From Finite to System Skyrme Interactions η = (ρ n ρ p )/ρ expansion under n p symmetry ( ρn ρ p E(ρ n, ρ p ) = E 0 (ρ)+s(ρ) ρ S(ρ) = aa V + L ρ ρ ρ 0 Finite Nucleus Nucleon densities ρ p (r) & ρ n (r) Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 +a (N Z )2 a() +E mic a a? = a V a a a = a V a + 2/3 a S a a a () =? half-infinite matter ) 2

7 Symmetry Energy: From Finite to System Skyrme Interactions η = (ρ n ρ p )/ρ expansion under n p symmetry ( ρn ρ p E(ρ n, ρ p ) = E 0 (ρ)+s(ρ) ρ S(ρ) = aa V + L ρ ρ ρ 0 Finite Nucleus Nucleon densities ρ p (r) & ρ n (r) Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 +a (N Z )2 a() +E mic a a? = a V a a a = a V a + 2/3 a S a a a () =? half-infinite matter ) 2

8 Symmetry Energy: From Finite to System Skyrme Interactions η = (ρ n ρ p )/ρ expansion under n p symmetry ( ρn ρ p E(ρ n, ρ p ) = E 0 (ρ)+s(ρ) ρ S(ρ) = aa V + L ρ ρ ρ 0 Finite Nucleus Nucleon densities ρ p (r) & ρ n (r) Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 +a (N Z )2 a() +E mic a a? = a V a a a = a V a + 2/3 a S a a a () =? half-infinite matter ) 2

9 Symmetry Energy: From Finite to System Skyrme Interactions η = (ρ n ρ p )/ρ expansion under n p symmetry ( ρn ρ p E(ρ n, ρ p ) = E 0 (ρ)+s(ρ) ρ S(ρ) = aa V + L ρ ρ ρ 0 Finite Nucleus Nucleon densities ρ p (r) & ρ n (r) Bethe-Weizsäcker formula: E = a V + a S 2/3 Z 2 + a C 1/3 +a (N Z )2 a() +E mic a a? = a V a a a = a V a + 2/3 a S a a a () =? half-infinite matter ) 2

10 Nucleus as Capacitor for symmetry E = a v + a s 2/3 + a a (N Z )2 = E 0 () + a a() (N Z )2 Capacitor analogy Q N Z E = E 0 + Q2 2C C 2a a () symmetry chemical potential nalogy µ a = E (N Z ) = 2a a() (N Z ) V = Q C V µ a

11 Nucleus as Capacitor for symmetry E = a v + a s 2/3 + a a (N Z )2 = E 0 () + a a() (N Z )2 Capacitor analogy Q N Z E = E 0 + Q2 2C C 2a a () symmetry chemical potential nalogy µ a = E (N Z ) = 2a a() (N Z ) V = Q C V µ a

12 Nucleus as Capacitor for symmetry E = a v + a s 2/3 + a a (N Z )2 = E 0 () + a a() (N Z )2 Capacitor analogy Q N Z E = E 0 + Q2 2C C 2a a () symmetry chemical potential nalogy µ a = E (N Z ) = 2a a() (N Z ) V = Q C V µ a

13 Invariant Densities Net density ρ(r) = ρ n (r) + ρ p (r) is isoscalar weakly depends on (N Z ) for given. (Coulomb suppressed... ) ρ np (r) = ρ n (r) ρ p (r) isovector but ρ np (r)/(n Z ) isoscalar! /(N Z ) normalizing factor global... Similar local normalizing factor, in terms of intense quantities, 2a V a /µ a, where a V a S(ρ 0 ) symmetric density (formfactor for isovector density) defined: ρ a (r) = 2aV a µ a [ρ n (r) ρ p (r)] Normal matter ρ a = ρ 0. Both ρ(r) & ρ a (r) weakly depend on η! In any nucleus ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] where ρ(r) & ρ a (r) have universal features!

14 Invariant Densities Net density ρ(r) = ρ n (r) + ρ p (r) is isoscalar weakly depends on (N Z ) for given. (Coulomb suppressed... ) ρ np (r) = ρ n (r) ρ p (r) isovector but ρ np (r)/(n Z ) isoscalar! /(N Z ) normalizing factor global... Similar local normalizing factor, in terms of intense quantities, 2a V a /µ a, where a V a S(ρ 0 ) symmetric density (formfactor for isovector density) defined: ρ a (r) = 2aV a µ a [ρ n (r) ρ p (r)] Normal matter ρ a = ρ 0. Both ρ(r) & ρ a (r) weakly depend on η! In any nucleus ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] where ρ(r) & ρ a (r) have universal features!

15 Invariant Densities Net density ρ(r) = ρ n (r) + ρ p (r) is isoscalar weakly depends on (N Z ) for given. (Coulomb suppressed... ) ρ np (r) = ρ n (r) ρ p (r) isovector but ρ np (r)/(n Z ) isoscalar! /(N Z ) normalizing factor global... Similar local normalizing factor, in terms of intense quantities, 2a V a /µ a, where a V a S(ρ 0 ) symmetric density (formfactor for isovector density) defined: ρ a (r) = 2aV a µ a [ρ n (r) ρ p (r)] Normal matter ρ a = ρ 0. Both ρ(r) & ρ a (r) weakly depend on η! In any nucleus ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] where ρ(r) & ρ a (r) have universal features!

16 Invariant Densities Net density ρ(r) = ρ n (r) + ρ p (r) is isoscalar weakly depends on (N Z ) for given. (Coulomb suppressed... ) ρ np (r) = ρ n (r) ρ p (r) isovector but ρ np (r)/(n Z ) isoscalar! /(N Z ) normalizing factor global... Similar local normalizing factor, in terms of intense quantities, 2a V a /µ a, where a V a S(ρ 0 ) symmetric density (formfactor for isovector density) defined: ρ a (r) = 2aV a µ a [ρ n (r) ρ p (r)] Normal matter ρ a = ρ 0. Both ρ(r) & ρ a (r) weakly depend on η! In any nucleus ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] where ρ(r) & ρ a (r) have universal features!

17 Nuclear Densities ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] Net isoscalar density ρ usually parameterized w/fermi function 1 ρ(r) = 1 + exp ( ) with R = r r R 0 1/3 d Isovector density ρ a?? Related to a a () & to S(ρ)! a a () = 2(N Z ) = 2 dr ρ np = 1 µ a µ a aa V dr ρ a (r) In uniform matter µ a = E (N Z ) = [S(ρ) ρ2 np/ρ] = 2 S(ρ) ρ np ρ ρ np ρ a = 2aV a ρ np = av a ρ µ a S(ρ) = Hartree-Fock calculations of half-infinite nuclear matter

18 Nuclear Densities ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] Net isoscalar density ρ usually parameterized w/fermi function 1 ρ(r) = 1 + exp ( ) with R = r r R 0 1/3 d Isovector density ρ a?? Related to a a () & to S(ρ)! a a () = 2(N Z ) = 2 dr ρ np = 1 µ a µ a aa V dr ρ a (r) In uniform matter µ a = E (N Z ) = [S(ρ) ρ2 np/ρ] = 2 S(ρ) ρ np ρ ρ np ρ a = 2aV a ρ np = av a ρ µ a S(ρ) = Hartree-Fock calculations of half-infinite nuclear matter

19 Nuclear Densities ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] Net isoscalar density ρ usually parameterized w/fermi function 1 ρ(r) = 1 + exp ( ) with R = r r R 0 1/3 d Isovector density ρ a?? Related to a a () & to S(ρ)! a a () = 2(N Z ) = 2 dr ρ np = 1 µ a µ a aa V dr ρ a (r) In uniform matter µ a = E (N Z ) = [S(ρ) ρ2 np/ρ] = 2 S(ρ) ρ np ρ ρ np ρ a = 2aV a ρ np = av a ρ µ a S(ρ) = Hartree-Fock calculations of half-infinite nuclear matter

20 Nuclear Densities ρ n,p (r) = 1 2 [ µ a ρ(r) ± 2aa V ρ a (r) ] Net isoscalar density ρ usually parameterized w/fermi function 1 ρ(r) = 1 + exp ( ) with R = r r R 0 1/3 d Isovector density ρ a?? Related to a a () & to S(ρ)! a a () = 2(N Z ) = 2 dr ρ np = 1 µ a µ a aa V dr ρ a (r) In uniform matter µ a = E (N Z ) = [S(ρ) ρ2 np/ρ] = 2 S(ρ) ρ np ρ ρ np ρ a = 2aV a ρ np = av a ρ µ a S(ρ) = Hartree-Fock calculations of half-infinite nuclear matter

21 Half-Infinite Matter in Skyrme-Hartree-Fock To one side infinite uniform matter & vacuum to the other Wavefunctions: Φ(r) = φ(z) e ik r matter interior/exterior: φ(z) sin (k z z + δ(k)) φ(z) e κ(k)z Discretization in k-space. Set of 1D HF eqs solved using Numerov s method until self-consistency: d dz B(z) d ( ) dz φ(z) + B(z) k 2 + U(z) φ(z) = ɛ(k) φ(z) Before: Farine et al, NP338(80)86

22 Net & symmetric Densities from SHF Results for different Skyrme interactions in half-infinite matter. Net & asymmetric densities displaced relative to each other. s symmetry energy changes gradually, so does the displacement.

23 symmetry Dependence of Net Density Results for different asymmetries

24 symmetry Dependence of symmetric Density ρ a = 2aV a µ a (ρ n ρ p ) Results for different asymmetries

25 Sensitivity to S(ρ) For weakly nonuniform matter, expected asymmetric density ρ a = ρ a V a /S(ρ) symmetric density ρ a oscillates around the expectation down to ρ ρ 0 /4

26 WKB nalysis Semiclassical sin ( z 0 k z (z ) dz ) allowed z < z 0 wavefunction φ z k (z) exp ( z κ z (z ) dz ) forbidden z > z 0 z 0 Density from ρ α (z) = dk φ k α(z) 2 Findings: t z < z 0 ρ a (z) av a ρ ( ρ 2/3 1 + S(ρ) S(ρ) F) where F(z) sin (2k F (z 0 z)), describing Friedel oscillations around ρ/s, up to the classical return point z 0. t z < z 0 ρ α (z) exp ( 2m ɛ F α z ) Fall-off governed by separation energy ɛ F α

27 WKB nalysis Semiclassical sin ( z 0 k z (z ) dz ) allowed z < z 0 wavefunction φ z k (z) exp ( z κ z (z ) dz ) forbidden z > z 0 z 0 Density from ρ α (z) = dk φ k α(z) 2 Findings: t z < z 0 ρ a (z) av a ρ ( ρ 2/3 1 + S(ρ) S(ρ) F) where F(z) sin (2k F (z 0 z)), describing Friedel oscillations around ρ/s, up to the classical return point z 0. t z < z 0 ρ α (z) exp ( 2m ɛ F α z ) Fall-off governed by separation energy ɛ F α

28 WKB nalysis Semiclassical sin ( z 0 k z (z ) dz ) allowed z < z 0 wavefunction φ z k (z) exp ( z κ z (z ) dz ) forbidden z > z 0 z 0 Density from ρ α (z) = dk φ k α(z) 2 Findings: t z < z 0 ρ a (z) av a ρ ( ρ 2/3 1 + S(ρ) S(ρ) F) where F(z) sin (2k F (z 0 z)), describing Friedel oscillations around ρ/s, up to the classical return point z 0. t z < z 0 ρ α (z) exp ( 2m ɛ F α z ) Fall-off governed by separation energy ɛ F α

29 Density Tails Two Skyrme interactions + different asymmetries

30 a a () = 1 aa V where a S a a V a Symmetry Coefficient dr ρ a (r) = 1 a V a r 0 3 R a and R a is displacement of isovector and isoscalar surfaces. a V a dr ρ(r) + 1 a V a surface region dr (ρ a ρ)(r) + 2/3 aa S L aa S Correlation f/shf 1 a a () = 1 a V a + 1/3 a S a

31 Charge Invariance?a a ()? Conclusions on sym-energy details, following E-formula fits, interrelated with conclusions on other terms in the formula: asymmetry-dependent Coulomb, Wigner & pairing + asymmetry-independent, due to (N Z )/ - correlations along stability line (PD NP727(03)233)! Best would be to study the symmetry energy in isolation from the rest of E-formula! bsurd?! Charge invariance to rescue: lowest nuclear states characterized by different isospin values (T,T z ), T z = (Z N)/2. Nuclear energy scalar in isospin space: sym energy E a = a a () (N Z )2 = 4 a a () T 2 z E a = 4 a a () T 2 = 4 a a() T (T + 1)

32 Charge Invariance?a a ()? Conclusions on sym-energy details, following E-formula fits, interrelated with conclusions on other terms in the formula: asymmetry-dependent Coulomb, Wigner & pairing + asymmetry-independent, due to (N Z )/ - correlations along stability line (PD NP727(03)233)! Best would be to study the symmetry energy in isolation from the rest of E-formula! bsurd?! Charge invariance to rescue: lowest nuclear states characterized by different isospin values (T,T z ), T z = (Z N)/2. Nuclear energy scalar in isospin space: sym energy E a = a a () (N Z )2 = 4 a a () T 2 z E a = 4 a a () T 2 = 4 a a() T (T + 1)

33 Charge Invariance?a a ()? Conclusions on sym-energy details, following E-formula fits, interrelated with conclusions on other terms in the formula: asymmetry-dependent Coulomb, Wigner & pairing + asymmetry-independent, due to (N Z )/ - correlations along stability line (PD NP727(03)233)! Best would be to study the symmetry energy in isolation from the rest of E-formula! bsurd?! Charge invariance to rescue: lowest nuclear states characterized by different isospin values (T,T z ), T z = (Z N)/2. Nuclear energy scalar in isospin space: sym energy E a = a a () (N Z )2 = 4 a a () T 2 z E a = 4 a a () T 2 = 4 a a() T (T + 1)

34 a a () Nucleus-by-Nucleus T (T + 1) E a = 4 a a () In the ground state T takes on the lowest possible value T = T z = N Z /2. Through +1 most of the Wigner term absorbed. Formula generalized to the lowest state of a given T (e.g. Jänecke et al., NP728(03)23). Pairing depends on evenness of T.?Lowest state of a given T : isobaric analogue state (IS) of some neighboring nucleus ground-state. T z =-1 T z =0 T z =1 Study of changes in the symmetry term possible nucleus by nucleus T=1 T=0

35 IS Data nalysis In the same nucleus: E 2 (T 2 ) E 1 (T 1 ) = 4 a a { T2 (T 2 + 1) T 1 (T 1 + 1) } +E mic (T 2, T z ) E mic (T 2, T z ) a 1 a () = 4 T 2 E? = (a V a ) 1 + (a S a ) 1 1/3 Data: ntony et al. DNDT66(97)1 E mic : Koura et al., ProTheoPhys113(05)305 v Groote et al., tdatnucdattab17(76)418 Moller et al., tdatnucdattab59(95)185

36 a a vs Line: best fit at > 20.

37 Best a()-descriptions... some problems w/extracting a a() from SHF for finite nuclei

38 Symmetry-Energy Parameters a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV

39 Modified Binding Formula E = a V + a S 2/3 Z 2 + a C 1/3 + aa V 1 + 1/3 aa V /aa S Energy Formula Performance: Fit residuals f/light asymmetric nuclei, either following the Bethe-Weizsäcker formula (open symbols) or the modified formula with a V a /a S a = 2.8 imposed (closed), i.e. the same parameter No. (N Z ) 2

40 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

41 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

42 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

43 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

44 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

45 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

46 Conclusions In bulk limit, ρ n & ρ p describable in terms of isoscalar ρ & isovector ρ a, η-independent. Surfaces for ρ & ρ a displaced by fixed distance R a ; different diffusenesses. side from the Friedel (shell) oscillations, ρ a (ρ n ρ p ) ρ/s(ρ) at ρ ρ 0 /4; separation-energy fall-off at ρ ρ 0 /4. -variation of the symmetry coefficient a a () tied to the difference between ρ a and ρ. Modified E-formula. Mass-dependence of a a determined from IS. Parameter values: a V a = ( ) MeV, a S a = (9.5 12) MeV, L 95 MeV; stiff symmetry energy. Outlook: finite nuclei - Coulomb & shell effects, learning on finer S(ρ)-details from ρ p (r) Thanks: Jenny Lee

47 nalytic Skin Size r 2 1/2 n r 2 1/2 p r 2 1/2 = 6NZ nalytic Expression for Skin Size symmetry energy only Coulomb correction N Z 1 + 1/3 a S a /a V a a C 168aa V 5/3 N /3 a S a /a V a 1 + 1/3 a S a /a V a Formula (lines) vs Typel & Brown results (symbols) from nonrelativistic & relativistic mean-field calculations, PRC64(01)027302

48 nalytic Skin Size Skin Sizes for Sn & Pb Isotopes Lines - formula predictions, PD NP727(03)233 Favored ratio a V a /a S a 32.5/