1 III. Nuclear Physics that determines the properties of the Universe Part I: Nuclear Masses 1. Why are masses important ? 1. Energy generation nuclear reaction A + B C if m A +m B > m C then energy Q=(m A +m B -m C )c 2 is generated by reaction 2. Stability if there is a reaction A B + C with Q>0 ( or m A > m B + m C ) then decay of nucleus A is energetically possible. nucleus A might then not exist (at least not for a very long time) 3. Equilibria for a nuclear reaction in equilibrium abundances scale with e -Q (Saha equation) Masses become the dominant factor in determining the outcome of nucleosynthesis “Q-value” Q = Energy generated (>0) or consumed (<0) by reaction

2 2. Nucleons MassSpinCharge Proton MeV/c 2 1/2+1e Neutron MeV/c 2 1/20 size: ~1 fm 3. Nuclei a bunch of nucleons bound together create a potential for an additional : nucleons attract each other via the strong force ( range ~ 1 fm) neutron proton (or any other charged particle) V r R V r R Coulomb Barrier V c Potential … … Nucleons in a Box: Discrete energy levels in nucleus R ~ 1.3 x A 1/3 fm

3 4. Nuclear Masses and Binding Energy Energy that is released when a nucleus is assembled from neutrons and protons m p = proton mass, m n = neutron mass, m(Z,N) = mass of nucleus with Z,N B>0 With B the mass of the nucleus is determined. B is roughly ~A Most tables give atomic mass excess  in MeV: Masses are usually tabulated as atomic masses (so for 12 C:  =0) Nuclear Mass ~ 1 GeV/A Electron Mass 511 keV/Z Electron Binding Energy 13.6 eV (H) to 116 keV (K-shell U) / Z m = m nuc + Z m e + B e

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5 Can be understood in liquid drop mass model: (Weizaecker Formula) (assumes incompressible fluid (volume ~ A) and sharp surface) x 1 ee x 0 oe/eo x (-1) oo Volume Term Surface Term ~ surface area (Surface nucleons less bound) Coulomb term. Coulomb repulsion leads to reduction uniformly charged sphere has E=3/5 Q 2 /R Asymmetry term: Pauli principle to protons: symmetric filling of p,n potential boxes has lowest energy (ignore Coulomb) protons neutrons protons lower total energy = more bound Pairing term: even number of like nucleons favoured (e=even, o=odd referring to Z, N respectively)

6 Best fit values (from A.H. Wapstra, Handbuch der Physik 38 (1958) 1) in MeV/c 2 aVaV aSaS aCaC aAaA aPaP Deviation (in MeV) to experimental masses: something is missing ! (Bertulani & Schechter)

7 Shell model: (single nucleon energy levels) Magic numbers are not evenly spaced shell gaps more bound than average less bound than average need to add shell correction term S(Z,N)

8 Understanding the B/A curve: neglect asymmetry term (assume reasonable asymmetry) neglect pairing and shell correction - want to understand average behaviour then ~surface/volume ratio favours large nuclei Coulomb repulsion has long range - the more protons the more repulsion favours small (low Z) nuclei const as strong force has short range maximum around ~Fe

9 5. Decay - energetics and decay law Decay of A in B and C is possible if reaction A B+C has positive Q-value BUT: there might be a barrier that prolongs the lifetime Decay is described by quantum mechanics and is a pure random process, with a constant probability for the decay to happen in a given time interval. N: Number of nuclei A (Parent) : decay rate (decays per second and parent nucleus) therefore lifetime  half-life T 1/2 =  ln2 = ln2/ is time for half of the nuclei present to decay

10 6. Decay modes for anything other than a neutron (or a neutrino) emitted from the nucleus there is a Coulomb barrier V r R Coulomb Barrier V c Potential unbound particle Example: for 197 Au -> 58 Fe I has Q ~ 100 MeV ! yet, gold is stable. If that barrier delays the decay beyond the lifetime of the universe (~ 14 Gyr) we consider the nucleus as being stable. not all decays that are energetically possible happen most common:  decay n decay p decay  decay fission

 decay (rates later …) p n conversion within a nucleus via weak interaction Modes (for a proton/neutron in a nucleus):  + decay electron capture  - decay p n + e + + e e - + p n + e n p + e - + e Electron capture (or EC) of atomic electrons or, in astrophysics, of electrons in the surrounding plasma Q-values for decay of nucleus (Z,N) Q  + / c 2 = m nuc (Z,N) - m nuc (Z-1,N+1) - m e = m(Z,N) - m(Z-1,N+1) - 2m e Q EC / c 2 = m nuc (Z,N) - m nuc (Z-1,N+1) + m e = m(Z,N) - m(Z-1,N+1) Q  - / c 2 = m nuc (Z,N) - m nuc (Z+1,N-1) - m e = m(Z,N) - m(Z+1,N-1) with nuclear masses with atomic masses Note: Q EC > Q  by MeV

12 Note: Q-values for reactions that conserve the number of nucleons can also be calculated directly using the tabulated  values instead of the masses Q-values with  values Q-values with binding energies B Q-values for reactions that conserve proton number and neutron number can be calculated using -B instead of the masses Example: 14 C -> 15 N + e +  (for atomic  ’s)

13  decay basically no barrier -> if energetically possible it usually happens (except if another decay mode dominates) therefore: any nucleus with a given mass number A will be converted into the most stable proton/neutron combination with mass number A by  decays valley of stability (Bertulani & Schechter)

14 Typical part of the chart of nuclides Z N blue: neutron excess undergo   decay red: proton excess undergo   decay even A isobaric chain odd A isobaric chain

15 Typical  decay half-lives: very near “stability” : occasionally Mio’s of years or longer more common within a few nuclei of stability: minutes - days most exotic nuclei that can be formed: ~ milliseconds

Neutron decay When adding neutrons to a nucleus eventually the gain in binding energy due to the Volume term is exceeded by the loss due to the growing asymmetry term then no more neutrons can be bound, the neutron drip line is reached Neutron drip line: S n = 0 beyond the neutron drip line, neutron decay occurs: (Z,N) (Z,N-1) + n Q-value: Q n = m(Z,N) - m(Z,N-1) - m n (same for atomic and nuclear masses !) beyond the drip line S n <0 Neutron Separation Energy S n S n (Z,N) = m(Z,N  1) + m n  m(Z,N) =  Q n for n-decay As there is no Coulombbarrier, and n-decay is governed by the strong force, for our purposes the decay is immediate and dominates all other possible decay modes the nuclei are neutron unbound Neutron drip line very closely resembles the border of nuclear existence !

17 Example: Neutron Separation Energies for Z=40 (Zirconium) neutron drip valley of stability add 37 neutrons

Proton decay same for protons … Proton drip line: S p = 0 beyond the drip line S n <0 Proton Separation Energy S p S p (Z,N) = m(Z  1,N) + m p  m(Z,N) the nuclei are neutron unbound

19 N=40 isotonic chain: add 7protons

20 Main difference to neutron drip line: When adding protons, asymmetry AND Coulomb term reduce the binding therefore steeper drop of proton separation energy - drip line reached much sooner Coulomb barrier (and Angular momentum barrier) can stabilize decay, especially for higher Z nuclei (lets say > Z~50) Nuclei beyond (not too far beyond) can therefore have other decay modes than p-decay. One has to go several steps beyond the proton drip line before nuclei cease to exist (how far depends on absolute value of Z). Note: have to go beyond Z=50 to prolong half-lives of proton emitters so that they can be studied in the laboratory (microseconds) for Z<50 nuclei beyond the drip line are very fast proton emitters (if they exist at all)

21 Example: Proton separation energies of Lu (Z=71) isotopes 35ms 80ms p-emitter EC/  decay  -emitter stabilizing effects of barriers slow down p-emission

22 neutrons protons Mass known Half-life known nothing known H(1) Fe (26) Sn (50) Pb (82) neutron dripline proton dripline note odd-even effect in drip line ! (p-drip: even Z more bound - can take away more n’s) (n-drip: even N more bound - can take away more p’s)

 decay emission of an  particle (= 4 He nucleus) Coulomb barrier twice as high as for p emission, but exceptionally strong bound, so larger Q-value emission of other nuclei does not play a role (but see fission !) because of increased Coulomb barrier reduced cluster probability <0, but closer to 0 with larger A,Z Q-value for  decay: large A therefore favored

24 lightest  emitter: 144 Nd (Z=60)(Q  MeV but still T 1/2 =2.3 x yr) beyond Bi  emission ends the valley of stability ! yellow are  emitter the higher the Q-value the easier the Coulomb barrier can be overcome (Penetrability ~ ) and the shorter the  -decay half-lives

Fission Very heavy nuclei can fission into two parts For large nuclei surface energy less important - large deformations less prohibitive. Then, with a small amount of additional energy (Fission barrier) nucleus can be deformed sufficiently so that coulomb repulsion wins over nucleon-nucleon attraction and nucleus fissions. (Q>0 if heavier than ~iron already) Separation (from Meyer-Kuckuk, Kernphysik)

26 Fission barrier depends on how shape is changed (obviously, for example. it is favourable to form a neck). Real theories have many more shape parameters - the fission barrier is then a landscape with mountains and valleys in this parameter space. The minimum energy needed for fission along the optimum valley is “the fission barrier” Real fission barriers: Example for parametrization in Moller et al. Nature 409 (2001) 485

27 Fission fragments: Naively splitting in half favourable (symmetric fission) There is a asymmetric fission mode due to shell effects (somewhat larger or smaller fragment than exact half might be favoured if more bound due to magic neutron or proton number) Both modes occur Example from Moller et al. Nature 409 (2001) 485

28 If fission barrier is low enough spontaneous fission can occur as a decay mode green = spontaneous fission spontaneous fission is the limit of existence for heavy nuclei

29 1. Nuclei found in nature are the stable ones (right balance of protons and neutron, and not too heavy to undergo alpha decay or fission) 2. There are many more unstable nuclei that can exist for short times after their creation (everything between the proton and neutron drip lines and below spont. Fission line) 3. Alpha particle is favored building block because explains peaks at nuclei composed of multiples of alpha particles in solar system abundance distribution Understanding some solar abundance features from nuclear masses: comparably low Coulomb barrier when fused with other nuclei high binding energy per nucleon - reactions that use alphas have larger Q-values 4. Binding energy per nucleon has a maximum in the Iron Nickel region in equilibrium these nuclei would be favored iron peak in solar abundance distribution indicates that some fraction of matter was brought into equilibrium (supernovae !)

30 1. ALL stable nuclei (and even all that we know that have half-lives of the order of the age of the universe, such as 238U or 232Th) are found in nature Explanation ? 2. Nucleons have maximum binding in 62Ni. But the universe is not just made out of 62Ni ! Explanation ? Surprises: