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SEMI-‐EMPIRICAL: LIQUID DROP MODEL AND NUCLEAR FISSION Remember… • Satura;on property of nuclear (strong) force • Rela;ve incompressibility of nuclear maIer (V α A) ! liquid drop model descrip;on (first model to describe nuclear proper;es) Liquid Drop Model • simplest approxima;on – nucleus is liquid drop with: R(µ,Á) = R0 (sphere) • allow for deforma;on: R(µ, Á) = R0(1+∑¸µ®¸µY¸µ*(µ, Á)) • if ®¸µ = ®¸µ(t) in some oscillatory approxima;on, derive collec;ve dynamics i.e. vibra;onal modes of nucleus – liquid drop is o[en star;ng point for collec;ve descrip;ons Charged Liquid Drop (1) Volume: binding of interior nucleons (bulk contribu;on) BE(A,Z) = aVA (2) Surface: fewer neighbours at surface BE(A,Z) = aVA – aSA2/3 (3) Coulomb: protons are charged and repel each other BE(A,Z) = aVA – aSA2/3 – aCZ(Z-‐1)A-‐1/3 Nucleons are not atoms (1) Symmetry energy: minimize energy based on occupied nucleon orbitals 2 -‐1 2/3 -‐1/3 BE(A,Z) = aVA – aSA – aCZ(Z-‐1)A – asym(A-‐2Z) A (2) Pairing: nucleons like to pair to 0+ 2 -‐1 BE(A,Z) = aVA – aSA2/3 – aCZ(Z-‐1)A-‐1/3 – asym(A-‐2Z) A + Pair where Pair = +± for even-‐even = 0 for even-‐odd (odd-‐A) = -‐± for odd-‐odd Pairing Term – 12 MeV/A1/2 Bethe-‐Weizsacher Mass Equa;on A2/3 – a Z(Z-‐1)A-‐1/3 – a 2 -‐1 BE(A,Z) = aVA – aS C sym(A-‐2Z) A + Pair where Pair = +± for even-‐even = 0 for even-‐odd (odd-‐A) = -‐± for odd-‐odd From a large-‐scale fit to available mass data: aV = 15.85 MeV aS = 18.34 MeV aC = 0.71 MeV asym = 23.21 MeV aP = 12 MeV, where ± = aP/A1/2 The LDM Nuclear Chart LDM: What’s Missing? LDM – What’s Missing? Fission • 1938: third decay mode discovered, by Hahn and Strassmann • bombard 92U with neutrons, expec;ng transuranium elements – found 56Ba • borrowing from biology, Meitner and Frisch coined the term fission • break-‐up of nucleus into smaller fragments – compe;;on between Coulomb and surface tension – natural in liquid-‐drop formalism Basic Fission Considera;ons • fission: spontaneous and neutron-‐induced • spontaneous fission: T1/2 from 1 ms to 1017 years – drama;c varia;on with Z • characteris;c mass distribu;ons – more asymmetric with higher mass, lower excita;on energy • experimentally accessible quan;ty: fission barrier height Fragment Mass Distribu;ons Fission Barrier Fission: Barrier Penetra;on Problem • spontaneous fission is a barrier penetra;on problem – barrier height usually » 6 MeV • barrier itself more complex than ® decay – realis;c descrip;on is double-‐humped • energy in fission: • a[er barrier penetra;on, scission point where fragments separate is reached, then fragments accelerated by Coulomb repulsion • energy predominantly in KE of fragments + separa;on energy and KE of neutrons + gamma-‐rays de-‐exci;ng fragments Fission Energe;cs Average Neutrons In Fission Deforma;on in Liquid Drop: Fission • deforma;on affects surface and Coulomb terms • deforma;on energy (I = (N-‐Z)/A): • calculate energy for series of deforma;ons – describe evolu;on during fission Shape Parameteriza;on • R(µ, Á) = R®(1+∑¸µ®¸µY¸µ*(µ, Á)) • R(µ, Á) must be real ! a¸µ* = (-‐1)µ®¸-‐µ • deformed radius related to spherical by volume constraint: • with expansion in R®, we obtain expressions for ES(d) and EC(d), to second order in ® Shape Parameteriza;on cont. • proper descrip;on of fission barrier requires at least third order in ® (deforma;on parameters) • simplify using symmetry of fission – assume rota;onal symmetry about deforma;on axis (defined as z-‐axis) – expansion now in P¸(cosµ) R(µ) = R¯(1+((2¸ + 1)/(4¼))1/2 ∑¸¯¸P¸(cosµ) • ¯2 most important at small deforma;on ! simplify with only ¯2 ≠ 0 S;ll Shape Parameteriza;on… • R(µ,a2) = Ra(a2)(1 + a2P2(cosµ)) • with this expansion, calculate the surface area, and Coulomb energy (volume provides normaliza;on) S = 4¼ R02(1 + (2/5)a22 – (4/105)a23 + …) EC = EC0(1 – (1/5)a22 – (4/105)a23 + …) Fissility • Define x = EC0/(2ES0) • small a2 (near spherical) – first term dominates • x < 1: posi;ve curvature in a2, stable against deforma;on • x > 1: unstable against deforma;on • large a2 and x < 1 -‐-‐ ¢ E is nega;ve – fission barrier develops Fission barrier and Fissility • Es;mate for barrier: • a2 = 0 and a2 = 7(1-‐x)/(1+2x) • Ebarrier = (98/15)*((1-‐x)3/(1+2x)2)*ES0 • Fissility is “readiness” of nucleus to fission • x > 1 – no barrier to fission at all • In liquid drop: • Spontaneous fission (254Fm), x = 0.841 – low barrier (2.4 MeV) Barrier Penetra;on + Fission Half-‐life • Change of deforma;on parameter: • WKB approach to tunneling: near top of barrier, P = (1 + exp(K))-‐1 Fission Half-‐life • decay probability = nP • n = number of ;mes “hitng” barrier • vibra;onal frequency along deforma;on axis n = 1021 s-‐1 • P requires evalua;on of ‘ac;on integral’, K • Assume parabolic barrier with height S, centered at ² = ²S Fission Half-‐life cont. • Define !F = (C/B)1/2 • Find for parabolic poten;al barrier, K = 2¼S/~!F • es;mate for P follows – this is essen;ally the Hill-‐Wheeler formula used to analyze fission data – result in range of experimental ~!F around 500 keV, and S from 2-‐8 MeV • but s;ll too simple – this is single-‐peak barrier model with limited applicability Mul;-‐Dimensional Deforma;on • we have assumed B(²) only • consider B®i®j(®1, ®2, …) – iner;al tensor • ac;on integral K then calculated over a trajectory in deforma;on space: • calculate and find Lmin – dynamical calcula;ons • calculate K(Lmin) to es;mate life;me Paths for Fission Fission Half-‐Lives