Symmetry Energy in Nuclear Surface Pawel Danielewicz Natl - PowerPoint PPT Presentation

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Symmetry Energy in Nuclear Surface Pawel Danielewicz Natl Superconducting Cyclotron Lab, Michigan State U Workshop on Nuclear Symmetry Energy at Medium Energies Catania & Militello V.C., May 28-29, 2008 Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Charge Symmetry & Charge Invariance Charge symmetry: invariance of nuclear interactions under n ↔ p interchange An isoscalar quantity F does not change under n ↔ p interchange. Example: nuclear energy. Expansion in η = ( N − Z ) / A for smooth F , has even terms only: F ( η ) = F 0 + F 2 η 2 + F 4 η 4 + . . . An 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 3 + . . . Note: G /η = G 1 + G 3 η 2 + . . . . Charge invariance: invariance of nuclear interactions under rotations in n - p space Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Charge Symmetry & Charge Invariance Charge symmetry: invariance of nuclear interactions under n ↔ p interchange An isoscalar quantity F does not change under n ↔ p interchange. Example: nuclear energy. Expansion in η = ( N − Z ) / A for smooth F , has even terms only: F ( η ) = F 0 + F 2 η 2 + F 4 η 4 + . . . An 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 3 + . . . Note: G /η = G 1 + G 3 η 2 + . . . . Charge invariance: invariance of nuclear interactions under rotations in n - p space Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Charge Symmetry & Charge Invariance Charge symmetry: invariance of nuclear interactions under n ↔ p interchange An isoscalar quantity F does not change under n ↔ p interchange. Example: nuclear energy. Expansion in η = ( N − Z ) / A for smooth F , has even terms only: F ( η ) = F 0 + F 2 η 2 + F 4 η 4 + . . . An 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 3 + . . . Note: G /η = G 1 + G 3 η 2 + . . . . Charge invariance: invariance of nuclear interactions under rotations in n - p space Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Symmetry Energy: From Finite to ∞ System Skyrme Interactions η = ( ρ n − ρ p ) /ρ expansion under n ↔ p symmetry � 2 � ρ n − ρ p E ( ρ n , ρ p ) = E 0 ( ρ )+ S ( ρ ) ρ a + L ρ − ρ 0 S ( ρ ) = a V + . . . 3 ρ 0 Finite Nucleus Nucleon densities ρ p ( r ) & ρ n ( r ) Bethe-Weizsäcker formula: A 1 / 3 + a a ( A ) ( N − Z ) 2 Z 2 E = − a V A + a S A 2 / 3 + a C + E mic A ? a + A 2 / 3 = a V A A a a a a = a a ( A ) =? a a V a S a ⇒ half-infinite matter Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Symmetry Energy: From Finite to ∞ System Skyrme Interactions η = ( ρ n − ρ p ) /ρ expansion under n ↔ p symmetry � 2 � ρ n − ρ p E ( ρ n , ρ p ) = E 0 ( ρ )+ S ( ρ ) ρ a + L ρ − ρ 0 S ( ρ ) = a V + . . . 3 ρ 0 Finite Nucleus Nucleon densities ρ p ( r ) & ρ n ( r ) Bethe-Weizsäcker formula: A 1 / 3 + a a ( A ) ( N − Z ) 2 Z 2 E = − a V A + a S A 2 / 3 + a C + E mic A ? a + A 2 / 3 = a V A A a a a a = a a ( A ) =? a a V a S a ⇒ half-infinite matter Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Symmetry Energy: From Finite to ∞ System Skyrme Interactions η = ( ρ n − ρ p ) /ρ expansion under n ↔ p symmetry � 2 � ρ n − ρ p E ( ρ n , ρ p ) = E 0 ( ρ )+ S ( ρ ) ρ a + L ρ − ρ 0 S ( ρ ) = a V + . . . 3 ρ 0 Finite Nucleus Nucleon densities ρ p ( r ) & ρ n ( r ) Bethe-Weizsäcker formula: A 1 / 3 + a a ( A ) ( N − Z ) 2 Z 2 E = − a V A + a S A 2 / 3 + a C + E mic A ? a + A 2 / 3 = a V A A a a a a = a a ( A ) =? a a V a S a ⇒ half-infinite matter Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Symmetry Energy: From Finite to ∞ System Skyrme Interactions η = ( ρ n − ρ p ) /ρ expansion under n ↔ p symmetry � 2 � ρ n − ρ p E ( ρ n , ρ p ) = E 0 ( ρ )+ S ( ρ ) ρ a + L ρ − ρ 0 S ( ρ ) = a V + . . . 3 ρ 0 Finite Nucleus Nucleon densities ρ p ( r ) & ρ n ( r ) Bethe-Weizsäcker formula: A 1 / 3 + a a ( A ) ( N − Z ) 2 Z 2 E = − a V A + a S A 2 / 3 + a C + E mic A ? a + A 2 / 3 = a V A A a a a a = a a ( A ) =? a a V a S a ⇒ half-infinite matter Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Symmetry Energy: From Finite to ∞ System Skyrme Interactions η = ( ρ n − ρ p ) /ρ expansion under n ↔ p symmetry � 2 � ρ n − ρ p E ( ρ n , ρ p ) = E 0 ( ρ )+ S ( ρ ) ρ a + L ρ − ρ 0 S ( ρ ) = a V + . . . 3 ρ 0 Finite Nucleus Nucleon densities ρ p ( r ) & ρ n ( r ) Bethe-Weizsäcker formula: A 1 / 3 + a a ( A ) ( N − Z ) 2 Z 2 E = − a V A + a S A 2 / 3 + a C + E mic A ? a + A 2 / 3 = a V A A a a a a = a a ( A ) =? a a V a S a ⇒ half-infinite matter Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Nucleus as Capacitor for Asymmetry E = − a v A + a s A 2 / 3 + a a A ( N − Z ) 2 = E 0 ( A ) + a a ( A ) ( N − Z ) 2 A Capacitor analogy  Q ≡ N − Z E = E 0 + Q 2  2 C ⇒ A C ≡  2 a a ( A ) Asymmetry chemical potential ∂ ( N − Z ) = 2 a a ( A ) ∂ E µ a = ( N − Z ) A Analogy V = Q C ⇒ V ≡ µ a Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Nucleus as Capacitor for Asymmetry E = − a v A + a s A 2 / 3 + a a A ( N − Z ) 2 = E 0 ( A ) + a a ( A ) ( N − Z ) 2 A Capacitor analogy  Q ≡ N − Z E = E 0 + Q 2  2 C ⇒ A C ≡  2 a a ( A ) Asymmetry chemical potential ∂ ( N − Z ) = 2 a a ( A ) ∂ E µ a = ( N − Z ) A Analogy V = Q C ⇒ V ≡ µ a Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Nucleus as Capacitor for Asymmetry E = − a v A + a s A 2 / 3 + a a A ( N − Z ) 2 = E 0 ( A ) + a a ( A ) ( N − Z ) 2 A Capacitor analogy  Q ≡ N − Z E = E 0 + Q 2  2 C ⇒ A C ≡  2 a a ( A ) Asymmetry chemical potential ∂ ( N − Z ) = 2 a a ( A ) ∂ E µ a = ( N − Z ) A Analogy V = Q C ⇒ V ≡ µ a Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Invariant Densities Net density ρ ( r ) = ρ n ( r ) + ρ p ( r ) is isoscalar ⇒ weakly depends on ( N − Z ) for given A . (Coulomb suppressed. . . ) ρ np ( r ) = ρ n ( r ) − ρ p ( r ) isovector but A ρ np ( r ) / ( N − Z ) isoscalar! A / ( N − Z ) normalizing factor global. . . Similar local normalizing factor, in terms of intense quantities, 2 a V a /µ a , where a V a ≡ S ( ρ 0 ) Asymmetric density (formfactor for isovector density) defined: ρ a ( r ) = 2 a V a [ ρ n ( r ) − ρ p ( r )] µ a Normal matter ρ a = ρ 0 . Both ρ ( r ) & ρ a ( r ) weakly depend on η ! In any nucleus ρ n , p ( r ) = 1 ρ ( r ) ± µ a � � ρ a ( r ) 2 a V 2 a where ρ ( r ) & ρ a ( r ) have universal features! Symmetry Energy in Surface Pawel Danielewicz

Introduction Energy & Densities Half- ∞ Matter a a ( A ) from Data Conclusions Invariant Densities Net density ρ ( r ) = ρ n ( r ) + ρ p ( r ) is isoscalar ⇒ weakly depends on ( N − Z ) for given A . (Coulomb suppressed. . . ) ρ np ( r ) = ρ n ( r ) − ρ p ( r ) isovector but A ρ np ( r ) / ( N − Z ) isoscalar! A / ( N − Z ) normalizing factor global. . . Similar local normalizing factor, in terms of intense quantities, 2 a V a /µ a , where a V a ≡ S ( ρ 0 ) Asymmetric density (formfactor for isovector density) defined: ρ a ( r ) = 2 a V a [ ρ n ( r ) − ρ p ( r )] µ a Normal matter ρ a = ρ 0 . Both ρ ( r ) & ρ a ( r ) weakly depend on η ! In any nucleus ρ n , p ( r ) = 1 ρ ( r ) ± µ a � � ρ a ( r ) 2 a V 2 a where ρ ( r ) & ρ a ( r ) have universal features! Symmetry Energy in Surface Pawel Danielewicz