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Lecture 1: Light and Unbound Nuclei

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Lecture 1: Light and Unbound Nuclei Lecture 1: Light and Unbound Nuclei TALENT Course 6: Theory of exploring nuclear reaction experiments Sofia Quaglioni LLNL-PRES-776621 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE- AC52-07NA27344. Lawrence Livermore National Security, LLC Outline § Lecture 1: Light and unBound nuclei § Lecture 2: AB initio theory of nuclear forces and nuclear structure § Lecture 3-4: Ab initio many-Body theory of nuclear scattering and reactions § Lecture 5: Applications to exotic nuclei and nuclear astrophysics 3 LLNL-PRES-776621 Lecture 1: Content § What are light and unbound nuclei? § What is the role of light and unbound nuclei in the Cosmos and on Earth? 4 LLNL-PRES-776621 What are light and unbound nuclei? What is light? A = N+Z ≲ 20 protons (Z) protons neutrons (N) 6 LLNL-PRES-776621 Binding energy (BE) § The energy required to disintegrate a nucleus into its components 2 2 2 BE(Z,N) = Z mp c + N mn c – M(Z,N) c § StaBle nuclei exhiBit a consideraBly large Binding energy 7 LLNL-PRES-776621 bound, What is unbound? stable unbound bound, EC/!+ unstable 7 -5 10 <T1/2< 10 sec bound, !- unstable 7 -5 10 <T1/2< 10 sec 8 LLNL-PRES-776621 Separation energy energy required to separate particle(s) § The minimum amount of energy that must be supplied to a nucleus to remove one or more neutrons/protons San(Z,N) = BE(Z,N) – BE(Z,N-a) § Moving away from the line of stability By progressively adding protons or neutrons causes the separation energy to decrease (exotic nuclei) and eventually cross the zero value (unBound nuclei) 9 LLNL-PRES-776621 The case of 5He § 4He Vghtly Bound (BE = 28.30 MeV) § 5He is not Bound. Why?!? 10 LLNL-PRES-776621 The case of 5He 1) Pauli exclusion principle Z=2 N=3 ℓ=1 0p1/2 0p3/2 ℓ=0 0s1/2 s-shell is full, extra neutron must be in p shell 11 LLNL-PRES-776621 The case of 5He 2) Centrifugal Barrier Overall potential is attractive but not enough to bind the system 12 LLNL-PRES-776621 Equation (TDSE) ˆ ih¯  = H  . (1.2) ˆt| Í | Í An eigenstate of the Hamiltonian H with energy E, i.e. a solution to the Time-Independent Schr¨odinger Equation (TISE) UnBound nuclear systems,H  = resonancesE  (1.3) | Í | Í has the simple time evolution iE Â(t, r)=exp t Â(0, r). (1.4) 3≠ h¯ 4 With the energy E real, the exponential factor is just a phase and the probability Â(t, r) 2 of finding the particle at a given r is unchanged over | Solution| of Solution of time. However,time if we-dependent let the energy be complextime-independent Schrodinger Γ Schrodinger equation E = E i , equation (1.5) 0 ≠ 2 we get iE Γ 2 Γ Â(t, r) 2 = exp 0 t exp t Â(0, r) =exp t Â(0, r) 2, | | - ≠ h¯ A≠2¯h B - A≠h¯ B| | - 3 4 - 13 LLNL-PRES-776621 - - - - (1.6) - - V V Continuum Scattering states Resonance Bound r 0 states r Bound states (a) A closed quantum system (b) An open quantum system Figure 1.1: A closed and an open quantum system. The closed system has an infinite number of bound, localized states, whereas the open system has unbound scattering states and resonances in addition to a finite number of bound states. 2 Equation (TDSE) Equation (TDSE) ˆ Equation (TDSE) ˆ Equationih¯  =(TDSE)H  . ih¯ (1.2) = Hˆ  . (1.2) ˆt| Í | Í ˆ ˆt| Í ih¯ | ÂÍ = H  . (1.2) An eigenstate of the Hamiltonian HAnwith eigenstate energy E of, i.e.the Hamiltonian aih¯ solution = toHH the with. ˆt| energyÍ E| ,Í i.e. a solution(1.2) to the ˆt| Í | Í Time-Independent Schr¨odinger EquationTime-IndependentEquation(TISE)An(TDSE)Equation eigenstate Schr(TDSE)¨odinger of the Hamiltonian Equation (TISE)H with energy E, i.e. a solution to the An eigenstate ofTime-Independent the Hamiltonian SchrH with¨odingerˆ energy EquationEˆ, i.e.(TISE) a solution to the ih¯  =ih¯H ÂÂ. = H  . (1.2) (1.2) Time-IndependentH  = E  Schr¨odinger EquationH (TISE)Â(1.3)ˆt|=ÍE  ˆ|t|Í Í | Í (1.3) | Í | Í | Í H| ÂÍ = E  (1.3) has the simple time evolution UnBoundhasAn the eigenstate nuclear simpleAn time eigenstate of systems, the evolution HamiltonianH  of the= resonancesE Hamiltonian H with| Í energyH with| ÍE, energy i.e. a solutionE(1.3), i.e. a to solution the to the Time-IndependenthasTime-Independent the simple Schr time¨odinger| Í evolution Schr Equation| ¨odingerÍ (TISE) Equation (TISE) has the simpleiE time evolution iE Â(t, r)=exp t Â(0, r). Â(t, r)=expH(1.4) = EtHÂÂÂ(0,iEr=).E  (1.4)(1.3) (1.3) 3≠ h¯ 4 ÂiE(t,|3r≠)=expÍ h¯ 4| |Í Í t |Â(0Í , r). (1.4) Â(t, r)=exp t Â(0, r). ≠ h¯ (1.4) With the energy E real, the exponentialWithhas the the factor simple energyhas is the just timeE simplereal, a evolution phase the time≠ and exponential evolutionh¯ the factor3 is4 just a phase and the 2 With the energy2 E 3real, the4 exponential factor is just a phase and the probability Â(t, r) of findingWith the the particleprobability energy at aE givenÂreal,(t, r)r theisof unchanged exponential finding the over particle factoriE is at just a given aiE phaser is unchanged and the over | | § Energyprobability| E is a real|  (numBer:t, rÂ)(t,2 rof)=exp finding| ( t, r the )=exp |2 = particlet | (0 , r at) |.2t a given(0, r).r is unchanged(1.4) over(1.4) time. However, if we let the energyprobability betime. complex However,(t, r) 2 of if finding we| let the| energy particle be at complex a≠ givenh¯ r is≠ unchangedh¯ over | time.| However, if we let the energy3 be4 complex3 4 time. However,Γ if we let the energy be complex Γ With theWith energy theE energyreal, theE exponentialreal,(1.5) the exponential factor is just factor a phase is just and(1.5) a phase the and the E = E0 i , 2 E2 = E0 i , Γ ≠probability§ Energy2 probabilityE is (at, complexr) ofÂ( findingt, numBer:r) of theΓ finding particle≠E =2 theE0 at particle ai given, atr ais given unchangedr is unchanged over (1.5) over | | E| = E0| i , ≠ 2 (1.5) we get wetime. get However,time. if However, we let the if we energy≠ let2 the be energy complex be complex we get we get 2 ResonanceΓ 2 stateΓ 2 iE0 Γ 2 iEΓ 0 2 Γ 2 Γ 2 (1.5) (1.5)  t, r t E =decayingtE0 Ei, rexponentially=, E0 i , t  , r , Â(t, r) = exp t exp t Â((0, r)) ==expexp 2 t Âexp(0,iEr)0 , 2≠(0 2Γ) =exp≠ 2 (0 Γ) 2 | | - 3≠ h¯ 4 A≠2¯h B|2 |- Â(-t,iEr0)A3≠=h¯h¯expB|4 Γ A≠| t2¯hexpB T = t-ln2/Â(0GΓ, rA) ≠=exph¯ B| 2 t| Â(0, r) , - Â(t, r)we= getexp- |we- gett| exp- ≠t h¯Â(0, r) A=exp≠1/22¯h -B t -Â(0, r) A, ≠h¯ B| | - | | - - ≠- h¯ - A≠32¯h B (1.6)4 - - A≠h¯ B|- | (1.6) - - - 3 - 4 - - - - - - - - 2 2 14 (1.6) LLNL-PRES-776621 - - - - (1.6) - 2 2- iE0 iE0 Γ- Γ - Γ Γ 2 2 Â-(t, r) =Â(expt, r) = expt exp t -texpÂ(0, r) t =expÂ(0, r) =expt Â(0, r)t , Â(0, r) , | | | - |≠ h¯- ≠Ah¯≠2¯h B A≠2¯h- B A≠- h¯ B| A≠h¯| B| | - 3 - 4 3 4 - - V V V - - V - - - V - - V - (1.6) (1.6) Continuum V - - Continuum V - - Continuum Scattering Scattering Continuum Scattering states V VScattering states V V states Continuum states Continuum Resonance Resonance ScatteringResonanceScattering Bound r Bound Resonancer 0 r states0 states states Bound states Bound r r r 0 0 r states states ResonanceResonance r Bound Bound Bound Bound r r Bound states Bound states 0 0 states states states states r r Bound Bound states states (a) A closed quantum system (a)(b) A An closed open quantum quantum system system (b) An open quantum system (a) A closed quantum(a) system A closed quantum system(b) An open quantum(b) system An open quantum system Figure 1.1: A closed and an open quantumFigure system. 1.1: A The closed closed and system an open has quantum system. The closed system has Figure 1.1: A closedFigure and 1.1: an open A closed quantum and an system. open quantum The closed system. system The has closed system has an infinite number of bound, localized states,an(a) infinite A whereas closed number(a) quantum the A open closed of bound, system system quantum localized has system states,(b) whereas An open(b) the quantum An open open system system quantum has system unbound scattering states and resonancesan infiniteunbound in number addition scatteringan of infinite to bound, a finite states number localized number and of states, resonances bound, of whereas localized in addition the states, open to whereas system a finite has the number open of system has unbound scattering states and resonances in addition to a finite number of bound states. boundFigure states.unbound 1.1:Figure A closed scattering 1.1: and A closed an states open and and quantum an resonances open system.
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