Isospin symmetry breaking in mirror nuclei Silvia M. Leni Dipartimento di Fisica dell Università and INFN, Padova, Italy
Energy scales MED 0-00 kev
The Nuclear landscape 87 primordial isotopes exist in Nature ~6000 nuclei are bound (3000 are known) NZ
A cut view
Nuclear shapes and excitation modes Spherical Deformed collective motion non-collective motion
Changing deformation along the band Rotational band 4 + + 0 + Oblate Band termination all single spins are aligned R j 8 + 6 + 4 + J(ħ) Backbend j + 0 + 8 + E γ (kev) JR+j +j 6 + 4 + + 0 + Prolate Deformed nucleus
The physics at the yrast line Isospin symmetry
Outline What s isospin? Why isospin symmetry? How do we study isospin symmetry? Experimental methods Theoretical methods What do we learn from the data?
Background What s isospin? Why isospin symmetry?
Symmetries Symmetries help to understand Nature Examination of fundamental symmetries: a key questions in Physics conservation laws good quantum numbers Conserved quantities imply some underlying symmetry enjoyed by the interactions In Nuclear Physics symmetries help to understand the behaviour of matter
Symmetries in nuclear physics p n Isospin Symmetry: 93 Heisenberg SU() Spin-Isospin Symmetry: 936 Wigner SU(4) π + J 0 j ω π + J 0 π + J Seniority Pairing: 943 Racah Spherical Symmetry: 949 Mayer Nuclear Deformed Field (spontaneous symmetry breaking) Restore symm. rotational spectra: 95 Bohr-Mottelson SU(3) Dynamical Symmetry: 958 Elliott Interacting Boson Model (IBM dynamical symmetry): 974 Arima and Iachello Critical point symm. E(5), X(5). 000 F. Iachello K. Heyde
The nucleus: a unique laboratory Composed by two types of fermions differing only on its charge Strong interaction: largely independent of the charge Proton Neutron exchange symmetry Proton and neutron can be viewed as two alternative states of the same particle: the nucleon. The quantum number that characteries the two charge states is the isospin t proton t : π neutron t + : ν
Charge symmetry and Isospin 93 Heisenberg applies the Pauli matrices to the new problem of labeling the two alternative charge states of the nucleon. 937 Wigner: isotopic spin is a good quantum number to characterie isobaric multiplets. T A t i,i N Z N Z N + T Z Isobaric analogue multiplets Charge Symmetry: V pp V nn Charge Independence: V pp V nn V pn π ν
Two-nucleon system For a two-nucleon system, four different isospin states can exist: Triplet T T, T T, T T, T 0 + ( ) Singlet T0 T 0,T 0 ( ) Total wavefunction must be antisymmetric on exchange of all (spin, space, isospin) co-ordinates Ψ space Ψ spin Ψ isospin
Isobaric multiplets It is easy to extend the formalism to many-nucleon systems The isospin quantum number T directly couples together the two effects of charge symmetry/independence and the Pauli principle T A t N Z N Z N + T,i i All nuclear states can be classified by the isospin quantum number, T, which to a good approximation can be treated as a good quantum number. Since T<T is forbidden, states of a given T can only occur in a set of nuclei with T T,T-,...,-T isospin multiplet or isobaric multiplet. This set of states of the same T in such a multiplet are termed isobaric analogue states (IAS). Z
Energy difference of ground states in A7 Look at the difference in absolute mass/binding energy Nuclei of same number of particles A7.6 MeV 7 Mg 7 Al 0 4.8 MeV 7 Si Protons 3 4 Neutrons 5 4 3 pp-pairs 66 78 9 nn-pairs 05 9 78 np-pairs 80 8 8 Total 35 35 35 pn-pairs can exist in states NOT allowed for pp or nn pairs - PAULI PRINCIPLE 7 Mg is less bound than 7 Al even though it has less protons, but it has also less p-n T0 pairs T0 stronger than T R.F. Casten
Analog states in isospin doublets Mirror nuclei with T ±/ 7 3Al 4 7 4Si 3 Test of isospin symmetry stable 4.6s
T isobaric triplets The nucleus can be characteried by isospin quantum numbers which restrict the possible states in which the many-nucleon system can exist. Look at the isobaric triplet: Mg0 Na 0Ne We expect: MeV 5 T states low in energy in Mg and Ne T0 and T states in Na (NZ) Tests isospin independence 5 MeV 4 3 4 + 4 + 4 4 + 3 0 + + 0 + 4 0.693 + + 0 + 3 + Mg 0 0Ne Na + 0 + 0
T isobaric triplets We expect: T states low in energy in 0 Na and 0 F MeV.5 T0 and T states in 0 Ne (NZ) MeV.5 MeV.5.5 0.5 0 4 + 4 + 4 3 + 3 + + 3 + + 0.73 3-0 0 + + 0 Na9 0 9 4 + 0Ne F 4 MeV 3.5 T - T 0 T + 3.5.5 0.5 + 0 0 + 0.5 0 Courtesy M.A. Bentley T allowed T 0 allowed T allowed Tests isospin independence
Three nucleon-system,, + T T T T,, + 3 3 3 3 T T T T,, + 3 3 T T T T We can form a doublet T/ or a quadruplet T3/
T3/ Isobaric quadruplets: the spectra F 0Ne Na 9 0 Mg 9 T3/ T/ SMALL differences in excitation energy due mainly to Coulomb effects LARGE differences in mass/binding energy - also due mainly to Coulomb effects. Courtesy M.A. Bentley
Coulomb displacement energies (CDE) in A -45-50 Isobaric Analogue States Ground States -BE(MeV) -55-60 -65 A Isobars CDE -70 8 9 0 3 Z CDE : For any two members of a multiplet of isospin T, transformed through exchange of k protons for neutrons is given by ( CDE) M,T,T k M,T,T k np α, T,T M α + α + (α Isobaric analogue state, defined by A, T, J)
Isobaric Multiplet Mass Equation A simple relationship between mass and T for a set of analogue states (E.P.Wigner 957) H CI αtt E αt H CV H CV k 0 αtt Any two-body force can be written in the isospin space in terms of 3 components ( k ) Isoscalar: H CV (0) (V nn +V pp +V pn )/3 Isovector: Isotensor: (IMME) To derive it we start with eigenstates of a charge-invariant (CI) Hamiltonian The eigenvalues do not depend on T and analogue states are degenerate A charge-violating interaction lifts this degeneracy: H CV () V pp - V nn H CV () V pp + V nn - V pn
IMME () The total binding energy can be obtained as BE( α TT ) αtt H + The energy splitting in an isobaric multiplet can be obtained as: CI H CV αtt ΔBE( α TT ) αtt H CV αtt Let s apply the Wigner-Eckart theorem αtt H CV αtt ( ) k 0 T-T ( T k T ) ( k ) αt H αt T 0 T CV Calling M ( k ) αt H ( k ) CV αt we obtain
IMME (3) ΔBE( αtt ) + + M T + T ( 0 ) M ( ) T ( T + )( T + ) 3T T ( T + ) ( T ) T ( T + )( T + )( T + 3) M ( ) We can now write the total BE into the form (a, b and c independent on T ): ( TT ) a bt ct BE α + + Isoscalar, ~ 00 s MeV (includes the contribution of V CI ) Isovector (Vpp Vnn) ~ 3-5 MeV (~ A /3 ) Isotensor (V pp +V nn V np ) ~00-300 kev The shift of the BE (mass) in an isobaric multiplet depends quadratically on T
Tests of IMME IMME widely tested through fitting T3/ quadruplets. Coefficient of cubic term (dt 3 ) should be ero. 0.4 0.3 c 0. 0. d 0-0. -0. -0.3-0.4 b/0-0.5 Coefficients of IMME -0.6-0.7 0 0 A 0 30 40 MeV N. Auerbach Phys. Rep 98(983)73 Benenson & Kashy RMP5(979)57 IMME works beautifully! The major contribution comes from the Coulomb interaction Courtesy M.A. Bentley
The Coulomb contribution to IMME Consider the nucleus a uniform charged sphere, the Coulomb energy is: C C R ) Z Z( e E 5 3 ( ) ( ) + + 0 4 5 3 3 A T T A A A r e ( ) ( ) 3 3 3 0 0 0 5 3 5 3 4 5 3 A r e c A A r e b A A A r e a This crude estimate (dashed-line in the figure) has to be corrected by exchange terms (Pauli Principle), etc.
The Nolen-Schiffer Anomaly IMME works well, but reproducing magnitude of coefficients historical problem CDEα,T,T M,T,T M,T,T M np For two adjacent nuclei, α α + + b c T + + M ( ) np Nolen & Schiffer (969) calculated CDE for wide range of isobaric multiplets Used independent-particle models (with exchange term) Corrected for other phenomena (e.g. electromagnetic spin-orbit (coming soon), magnetic effects, core-polarisation, isospin impurity, etc.). But magnitude of predicted CDE ALWAYS ~5-8% lower than experimental values. Underestimated by ~500 kev! Auerbach (983) improves the theoretical description but the anomaly remained Duflo and Zuker (00) reduce the difference to ~00-00 kev
Coulomb Energy Differences (CED) What about excited states - Coulomb Energy Differences Normalise ground state energies and take differences in excitation energy (CED) How does Coulomb energy (+ other INC) change with E x, J? Does Nolen-Schiffer Anomaly only concern ground states? 9
CED: MED and TED Z Mirror Energy Differences N Ex Ex MED J J,T / J,T + / Δb Test the charge symmetry of the interaction J Triplet Energy Differences J ExJ,T xj,t x + E E + J,T TED 0 Δc Test the charge independency of the interaction J
A classical example: MED in T/ states Coulomb effects in isobaric multiplets: - bulk energy (00 s of MeV) - displacement energy (g.s.) CDE (0 s of MeV) - differences between excited states (0 s of kev) Mirror Energy Differences MEDJ EJ ( Z > N) EJ ( Z < N) 49 49 5Mn 4 4 Cr 5
Symmetry breaking Isospin symmetry breakdown, mainly due to the Coulomb field, manifests when comparing mirror nuclei. This constitutes an efficient observatory for a direct insight into nuclear structure properties.
Measuring the Isospin Symmetry Breaking We will see that CED are extremely sensitive to quite subtle nuclear structure phenomena which, with the aid of shell model calculations, can be interpreted quantitatively at the level of 0 s of kev. We measure nuclear structure features: How the nucleus generates its angular momentum Evolution of the radii (deformation) along a rotational band Learn about the configuration of the states Isospin non-conserving terms in the nuclear interaction
Understanding structure features 0 J (ħ) 5 0 5 0 3 4 5 E γ (MeV) Most of the structure features of nuclei in the f 7/ shell are very well described by shell model calculations in the full fp valence space What happens at the backbending? band-crossing? alignment? which nucleons are aligning? S.M.Leni et al., PRC 56, 33 (997)
Mirror energy differences and alignment J j j probability distribution for the relative distance of two like particles in the f7/ shell j j 8 ΔE C 8 j j 6 neutron align. 4 0 j j A(N,Z) A(Z,N) j 6 4 0 J6 proton align. j J0 J.A. Cameron et al., Phys. Lett. B 35, 39 (990) MED 0 I8 angular momentum Shifts between the excitation energies of the mirror pair at the back-bend indicate the type of nucleons that are aligning
Nucleon alignment at the backbending J.A. Cameron et al., Phys. Lett. B 35, 39 (990) C.D. O'Leary et al., Phys. Rev. Lett. 79, 4349 (997) j j 49 Μν 49 Χρ Experimental MED j j J6 Alignment MED are a probe of nuclear structure: reflect the way the nucleus generates its angular momentum
Nucleon alignment at the backbending D.D. Warner, M.A. Bentley and P. Van Isacker., Nature Physics (006) 3 Energy (MeV) 6 5 F e 5 Mn Experiment fp-shell Mod Experiment Shell Model 5 MED Alignment 3 7 J 3 9 0 00 0-00 MED (kev) 5
Cranked shell model and alignment alignment Cranked shell-model Approximations: one shell only fixed deformation no p-n pairing CSM: good qualitative description of the data J.A. Sheikh, P. Van Isacker, D.D. Warner and J.A. Cameron, Phys. Lett. B 5 (990) 34
Alignment and shell model Define the operator A π ( + ) J 6 a a ( a a ) + J 6 π π π π Counts the number of protons coupled to J6 ΔA Calculate the difference of the expectation value in both mirror as a function of the angular momentum A ( Z> ) Φ J Φ' J Aπ( Z ) Φ J π, J Φ J π < ' 7/ 5/ 3/ / 5 Fe 5 Mn π ν π ν In 5 Fe ( 5 Mn) a pair of protons (neutrons) align first and at higher frequency align the neutrons (protons) 0 Alignment 5 Fe- 5 Mn M.A.Bentley et al. Phys Rev. C6 (000) 05303
Alignment in odd- and even-mass nuclei In odd-mass nuclei, the type of nucleons that aligns first is determined by the blocking effect the even fluid will align first 7/ 5/ 3/ / 5 Fe 5 Mn π ν π ν What about even-even rotating nuclei? 50 Fe 50 Cr π ν π ν
Alignment in even-even rotating nuclei Shell Renormaliation model only qualitative of the Coulomb description m.e.? Only a Coulomb effect? calculate for protons in both mirrors: ΔA ( Z> ) Φ J Φ' J π ( Z ) Φ' J Φ A A π J π < Cr-Fe counts the number of aligned protons - ΔA π In 50 Cr ( 50 Fe) a pair of protons (neutrons) align first and at higher frequency align the neutrons (protons) can shell model do better? S.M.Leni et al., Phys. Rev. Lett. 87, 50 (00)
Bibliography ) Review article on CED: M.A. Bentley and S.M. Leni, Prog. Part. Nucl. Phys. 59 (007) 497-56. ) Isospin in Nuclear Physics, Ed. D.H. Wilkinson, North Holland, Amsterdam, 969. 3) Review article on CDE: J.A. Nolen and J.P. Schiffer, Ann. Rev. Nucl. Sci 9 (969) 47 4) N. Auerbach, Phys. Rep. 98 (983) 73 5) Theory on CDE: J. Duflo and A.P. Zuker, Phys. Rev. C 66 (00) 05304(R) 6) Theory on CED: A.P. Zuker, S.M. Leni, G. Martine-Pinedo and A. Poves, Phys. Rev. Lett. 89 (00) 450; 7) Theory on CED: J.A. Sheikh, D.D. Warner and P. Van Isacker, Phys. Lett. B 443 (998) 6 8) Shell model reviews: B.A. Brown, Prog. Part. Nucl. Phys 47 (00) 57; T. Otsuka, M. Honma, T. Miusaki and N. Shimitu, Prog. Part. Nucl. Phys 47 (00) 39; E. Caurier, G. Martine-Pinedo, F. Nowacki, A.Poves, and A.P. Zuker, Rev. Mod. Phys. 77 (005) 47 9) Review article on N~Z: D. D. Warner, M. A. Bentley, P. Van Isacker, Nature Physics, (006) 3-38
Lecture The end