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III. Nuclear Physics that determines the properties of the Universe

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## III. Nuclear Physics that determines the properties of the Universe

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**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 mA+mB > mC then energy Q=(mA+mB-mC)c2 is generated by reaction “Q-value” Q = Energy generated (>0) or consumed (<0) by reaction 2. Stability with Q>0 ( or mA> mB + mC) if there is a reaction A B + 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**…**… Nucleons in a Box:Discrete energy levels in nucleus 2. Nucleons size: ~1 fm 3. Nuclei nucleons attract each other via the strong force ( range ~ 1 fm) a bunch of nucleons bound together create a potential for an additional : neutron proton(or any other charged particle) V V Coulomb Barrier Vc R ~ 1.3 x A1/3 fm Potential Potential R R r r**4. Nuclear Masses and Binding Energy**Energy that is released when a nucleus is assembled from neutrons and protons mp = proton mass, mn = 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 Masses are usually tabulated as atomic masses m = mnuc + Z me + Be Nuclear Mass ~ 1 GeV/A Electron Binding Energy13.6 eV (H)to 116 keV (K-shell U) / Z Electron Mass 511 keV/Z Most tables give atomic mass excess D in MeV: (so for 12C: D=0)**Can be understood in liquid drop mass model: (Weizaecker**Formula) (assumes incompressible fluid (volume ~ A) and sharp surface) Volume Term Surface Term ~ surface area (Surface nucleons less bound) Coulomb term. Coulomb repulsion leads to reduction uniformly charged sphere has E=3/5 Q2/R Asymmetry term: Pauli principle to protons: symmetric filling of p,n potential boxes has lowest energy (ignore Coulomb) lower totalenergy =more bound neutrons protons neutrons protons x 1 ee x 0 oe/eo x (-1) oo Pairing term: even number of like nucleons favoured (e=even, o=odd referring to Z, N respectively)**Best fit values (from A.H. Wapstra, Handbuch der Physik 38**(1958) 1) in MeV/c2 Deviation (in MeV) to experimental masses: (Bertulani & Schechter) something is missing !**Shell model:(single nucleon energy levels)**are not evenly spaced shell gaps less boundthan average more boundthan average need to addshell correction termS(Z,N) Magic numbers**Understanding the B/A curve:**neglect asymmetry term (assume reasonable asymmetry) neglect pairing and shell correction - want to understand average behaviour then constas strong force hasshort range Coulomb repulsion has long range- the more protons the more repulsion favours small (low Z) nuclei ~surface/volume ratiofavours large nuclei maximum around ~Fe**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) l : decay rate (decays per second and parent nucleus) therefore lifetime t=1/l half-life T1/2 = t ln2 = ln2/l is time for half of the nuclei present to decay**6. Decay modes**for anything other than a neutron (or a neutrino) emitted from the nucleusthere is a Coulomb barrier V Coulomb Barrier Vc unboundparticle R r Potential If that barrier delays the decay beyond the lifetime of the universe (~ 14 Gyr)we consider the nucleus as being stable. Example: for 197Au -> 58Fe + 139I has Q ~ 100 MeV ! yet, gold is stable. not all decays that are energetically possible happen most common: • b decay • n decay • p decay • a decay • fission**n p + e- + ne**p n + e+ + ne e- + p n + ne (rates later …) 6.1. b decay p n conversion within a nucleus via weak interaction Modes (for a proton/neutron in a nucleus): b+ decay electron capture b- decay Electron capture (or EC) of atomic electrons or, in astrophysics, of electrons in the surrounding plasma Q-values for decay of nucleus (Z,N) with nuclear masses with atomic masses = m(Z,N) - m(Z-1,N+1) - 2me Qb+ / c2 = mnuc(Z,N) - mnuc(Z-1,N+1) - me QEC / c2 = mnuc(Z,N) - mnuc(Z-1,N+1) + me = m(Z,N) - m(Z-1,N+1) Qb- / c2 = mnuc(Z,N) - mnuc(Z+1,N-1) - me = m(Z,N) - m(Z+1,N-1) Note: QEC > Qb+ by 1.022 MeV**Q-values with D values**Note: Q-values for reactions that conserve the number of nucleons can also be calculated directly using the tabulated D values instead of the masses Example: 14C -> 15N + e +ne (for atomic D’s) 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**valley of stability**b 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 b decays (Bertulani & Schechter)**odd Aisobaric chain**even Aisobaric chain Typical part of the chart of nuclides red: proton excessundergo b+ decay blue: neutron excessundergo b- decay Z N**Typical b decay half-lives:**very near “stability” : occasionally Mio’s of years or longer more common within a few nuclei of stability: minutes - days ~ milliseconds most exotic nuclei that can be formed:**6.2. Neutron decay**When adding neutrons to a nucleus eventually the gain in binding energy dueto 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 beyond the neutron drip line, neutron decay occurs: (Z,N) (Z,N-1) + n Qn = m(Z,N) - m(Z,N-1) - mn Q-value: (same for atomic and nuclear masses !) Neutron Separation Energy Sn Sn(Z,N) = m(Z,N-1) + mn- m(Z,N) = -Qn for n-decay Neutron drip line: Sn= 0 beyond the drip line Sn<0 the nuclei are neutron unbound 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 Neutron drip line very closely resembles the border of nuclear existence !**Example: Neutron Separation Energies for Z=40 (Zirconium)**add 37 neutrons valley of stability neutron drip**6.3. Proton decay**same for protons … Proton Separation Energy Sp Sp(Z,N) = m(Z-1,N) + mp- m(Z,N) Proton drip line: Sp= 0 beyond the drip line Sn<0 the nuclei are neutron unbound**N=40 isotonic chain:**add 7protons**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)**p-emitter**EC/b decay a-emitter 80ms 35ms stabilizing effects of barriers slow down p-emission Example: Proton separation energies of Lu (Z=71) isotopes**Mass known**Half-life known nothing known Pb (82) Sn (50) proton dripline Fe (26) neutron 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) protons H(1) neutrons**6.4. a decay**emission of an a particle (= 4He 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 Q-value for a decay: <0, but closer to 0 with larger A,Z large A therefore favored**lightest a emitter: 144Nd (Z=60)**(Qa=1.9 MeV but still T1/2=2.3 x 1015 yr) beyond Bi a emission ends the valley of stability ! yelloware a emitter the higher the Q-value the easier the Coulomb barrier can be overcome(Penetrability ~ )and the shorter the a-decay half-lives**6.5. Fission**Very heavy nuclei can fission into two parts (Q>0 if heavier than ~iron already) For large nuclei surface energy less important - large deformations less prohibitive. Then, with a small amount of additional energy (Fission barrier) nucleuscan be deformed sufficiently so that coulomb repulsion wins over nucleon-nucleon attraction and nucleus fissions. Separation (from Meyer-Kuckuk, Kernphysik)**Real fission barriers:**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” Example for parametrization in Moller et al. Nature 409 (2001) 485**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 favouredif more bound due to magic neutron or proton number) Both modes occur Example from Moller et al. Nature 409 (2001) 485**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**Understanding some solar abundance features from nuclear**masses: 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 • comparably low Coulomb barrier when fused with other nuclei • high binding energy per nucleon - reactions that use alphas have larger Q-values explains peaks at nuclei composed of multiples of alpha particles in solar system abundance distribution 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 !)**Surprises:**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 ?