Download semi-‐empirical: liquid drop model and nuclear fission

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