1. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 1 CH.I : INTRODUCTION TO REACTOR PHYSICS FROM THE FISSION PROCESS TO THE REACTOR CHARACTERISTICS DEFINITION NUCLEAR FUELS n – heavy nucleus INTERACTION NEUTRON CYCLE AND CRITICALITY CONSTITUTIVE ELEMENTS OF REACTORS CROSS SECTIONS PROFILES INTERACTION MECHANISMS ASSOCIATED CROSS SECTIONS NUCLEAR FISSION DESCRIPTION DELAYED NEUTRONS

2. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 2 I.1 FROM THE FISSION PROCESS TO THE REACTOR CHARACTERISTICS DEFINITION NUCLEAR FUELS Energy production by nuclear fission Stability of heavy nuclides  Excess of neutrons (n) with respect to protons  Last stable element in natural conditions: U (Z=92) Natural distribution of U isotopes: U 28 (A=238) : ~ 99,3 % U 25 (A=235) : ~ 0,7% U 24 (A=234) : traces

3. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 3 NUMBER OF NEUTRONS (N) Stable nuclides – N-Z relation

4. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 Binding energy B(A,Z) of the nuclei Default of mass of a nucleus:  between  masses of its constituents and its own mass  Maximum about A~50  Possible release of energy either by fusion of light nuclei or fission of heavy nuclei Yet a spontaneous process? 4

5. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 5 Fissile and fertile isotopes Nuclear reactor  possibility to fission nuclei –by neutron bombing, –with production of additional neutrons, –and chain reaction self-sustained Few possible isotopes (called fissile): U 235 (natural), U 233, Pu 239 Fertile isotopes: neutron capture  fissile isotopes U 238 + n  U 239 +   Np 239  Pu 239 Th 232 + n  Th 233 +   Pa 233  U 233 -- -- -- -- 2.3 j 27.4 j22’ 23’

6. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 6 Fission energy (same order of magnitude for other fissile isotopes) 1g U 235  200 MeV x N A / 235  8.2 x 10 10 J  ~ 1 MWj ! (some losses however + U 235 far from being totally consumed in current PWRs) fission combustion 1 nucleus U 235 200 MeV 1 atom C3 eV

7. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 7 Thermal power – MW thermal (MWth) Energy produced by the reactor / unit time Electrical power – MW electric (MWe) output of the alternator Efficiency of the thermodynamic cycle: ~ 33% Power of PWR reactors: 900 (3 loops) or 1300 (4 l.) Mwe  Consumption of fissile material / day = … BUT: natural abundance of U isotopes unfavorable!  Artificial enrichment of the nuclear fuel in U 235 to reach a sufficient power density (for light water reactors)

8. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 8 ‘n – heavy nucleus’ INTERACTIONS Possible phenomena Scattering due to collisions Fission: U 25 + n  2 fragments + n +  and  radiations ( : nb of n / fission) Radiative capture: U 25 + n  U 26 +  … elastic inelastic

9. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 9 Cross section Proba of those interactions? Macroscopic cross section  interaction proba of a n per unit length of its free flight in a media  : [cm -1 ] function of  the media  the energy of the n (i.e. relative velocity v between n – nucleus)  the spatial position (if variation in the isotopic composition of the material)  the interaction type

10. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 10 Cross sections considered in reactor physics  in Inelastic scattering ee Elastic scattering  s =  e +  in Total scattering ff Fission cc  capture  a =  c +  f Absorption (capture + fission)  t =  a +  s All interactions (Q: type of proba density function (pdf) for the free flight?) For an isotope,   N, isotopic density ([cm -3 ])   = N.  f(p,T) f(nucleus, v)  : microscopic cross section [cm 2 ]

11. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 11 Illustrative interpretation of  : ‘visible’ cross section of a nucleus for a n of velocity v  More appropriate unit: 1 barn = 10 -24 cm 2 Dependence of  as a function of energy E (  proba of a particular interaction) See next slides Energy ranges to be accounted for:  Energy of the n produced by fission: O(MeV)  Energy of the n after their slowing-down due to successive collisions, i.e. thermal (thermal equilibrium with the media) : 0.025 eV  8 decades !

12. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 12 See chap. VI

13. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 13

14. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 14 http://www.world-nuclear.org/education/phys.htm

15. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 15 Behavior of  f (U 235 )  Fissions much more probable at low energy  Numerous resonances at intermediate energies  Thermalisation of the n  Efficient slowing-down of the n (with a material with a low A, e.g. H 2 )  Absorptions avoided (low capture proba)  Possible moderator : H 2 0 (also cooling fluid)  Thermal reactors

16. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 16 Behavior of  f (U 238 )  Fissile at high energy (> 0.8 MeV)   f (U 238 ) <<  f (U 235 ) But possible production of Pu 239, fissile with a high Breeding ratio: BR = BR > 1: conversion  breeding  Interest of fast reactors But: n slowing-down to be avoided  Coolant: molten metals (Na!), molten salts… Production rate of fissile material Destruction rate of fissile material (reconsidered in Gen-IV)

17. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 17 NEUTRON CYCLE AND CRITICALITY Cycle characteristics 1. Expected number of n / fission: (E) For U 235 : (E in MeV)

18. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 18 2.Regeneration factor with  =  c /  f Strong dependence on E !  For U 235 thermal n:  = 2.07  For U nat thermal n:  = 1.34  of  in the fast domain because   Mixture of isotopes: Nb n produced in the fuel Nb n absorbed in the fuel  =

19. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 19 3.Breeding ratio: BR = (  - 1).F if F = fraction of the fission n absorbed in fertile material because  - 1 : excess of n available for breeding Distribution of the reaction products n + fissile isotope  / (1 +  ) isotope resulting from capture (e.g. U 236 for U 235 ) + 2 / (1 +  ) fission products +  n (+ ,  radiations)

20. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 20 Criticality Reactor in regime mode  nb n produced = nb n consumed  chain reaction exactly self-sustained Multiplication factor of a reactor: k = expected nb of n emitted during a cycle/initial n  Rules the evolution of the neutron population:  k < 1 : sub-critical reactor  k = 1 : critical reactor  k > 1 : super-critical reactor

21. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 21 Expression of the multiplication factor (case of a thermal reactor with uranium) Infinite reactor: k  fission 1 thermal n absorbed in the fuel E O(MeV) 0.025 eV  fission n (fast)  n Fast fissions in U 238  : fast fission factor p : resonance escape probability f : thermal usage factor Absorption of the thermal n in the fuel Slowing-down  p n  pf n k  =  pf Typical values:  = 1.65  = 1.02 p = 0.87f = 0.71 0.8MeV

22. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 22 Finite reactor: k eff leakages !!! P f : non-leakage proba during slowing-down P th : non-leakage proba at thermal energy P = P f P th 1 P, k  : f(geometry and nature of materials, enrichment) But leakages dependent on the surface/volume ratio  Critical size (hence mass) of the reactor  Balance between geometry and neutronics properties Leakage reduction: reflector around the reactor Reactivity: “distance to criticality” Main protection: neutron-absorbing control rods Emergency shutdown (“scram”): control rods dropped between fuel rods k eff = k .P f P th =  pfP f P th

23. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 CONSTITUTIVE ELEMENTS OF REACTORS 23 ElementObjectiveExample Fissile isotopesProvide the energy from fissions U 235, U 233, Pu 239 Fertile isotopesConvertible in fissile isotopesU 238, Th 232 Fission energyThermal  fastThermal reactors  breeders (breeders) ModeratorSlow down fission n in thermal n H 2 O, D 2 O, graphite CoolantCool down the core and transport the energy produced by fissions H 2 O, D 2 O (thermal) Na, molten metals (fast) Thermodynamic cycle Pressurized (PWR) or boiling (BWR) water

24. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 24 ElementObjectiveExample n-absorbing materials Control reactivityControl rods ReflectorLimit leakages (located around the core, same properties as moderator) ShieldingReduce losses and dosesThermal and biological walls … Many reactor designs were proposed Economical interest of some of them only (PWR, BWR…) Technological difficulties met for some designs (see Na technology for fast breeders when first tried) New or more developed designs under study

25. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 25 I.2 CROSS SECTION PROFILES INTERACTION MECHANISMS associated to the n  nucleus size (in the E range considered)  Interaction with the nucleus as a whole Two mechanisms:  Scattering of potential : « shocks » without interaction with the internal structure of the nucleus  Resonances  interaction with the internal structure:  Constitution of a composed nucleus  Deexcitation E excitation = f(E binding (n),E kinetic (n))

26. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 Energy carried by an incident n on a nucleus System 1: n incident (with E cin ) on a (A,Z) nucleus  Total energy of the system: E tot =m (A,Z) c 2 +m n c 2 +E cin System 2: Compound nucleus (A+1,Z)  Total energy of the system: E tot =m (A+1,Z) c 2 +E cin +…? Binding energy of nucleus (A,Z)  Energy associated to the mass default of the nucleus: B(A,Z)=(A-Z)m n c 2 +Zm p c 2 -m (A,Z) c 2 Separation energy of a n out of nucleus (A+1,Z)  Energy to be provided to the n to escape from the nucleus: S n =B(A+1,Z)-B(A,Z)=m n c 2 +m (A,Z) c 2 -m (A+1,Z) c 2 Therefore: E tot =m (A+1,Z) c 2 +E cin +S n  As if the incident n was bringing E cin + S n to the compound nucleus 26

27. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 27 ASSOCIATED CROSS SECTIONS  capture n absorbed composed nucleus     : resonances at the E levels of the composed kernel  Breit-Wigner at level E o (for well-separated levels) k : wave nb  k 2  E (relative E kin of the n – nucleus system) g J : statistical spin factor  ,  n,  (=   +  n ) : peak width (capture, scattering and total, resp.) deexcitation x = (E-E o ) / (  /2)

28. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 28 Behavior of     indep. of E and  n   E E<<E o :   ~ 1 /  E  region in 1/v E>>E o :   ~ 1 / E 5/2       max =  o.(   /  ) ~ 1 /  E o  resonances smaller and smaller and less and less separated

29. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 29 Scattering Scattering of potential Elastic scattering of the n without composed kernel   pa = 4  R 2 for E  [1eV,1MeV] (R = radius of the nucleus) Elastic scattering of resonance n absorbed composed nucleus n reemitted with “the same” E   s : similar shape to a capture resonance + scattering of potential + interference interference (i.e. E of the c.o.m. conserved !!)

30. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 30

31. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 31 Inelastic scattering n absorbed composed nucleus n reemitted at E < E i + deexcitation of the nucleus by  emission Condition: n with sufficient E i to excite the 1 st level of the nucleus   in = 0 up to a threshold (~ 10 keV for heavy nuclei) Rem: threshold much higher for light nuclei  not involved in the inelastic scattering taking place in a reactor  What about moderation by light nuclei (e.g. H 2 ) ? By elastic collisions in the c.o.m. (center-of-mass) system !!  Relative E conserved  Efficient slowing-down of the n

32. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 32 Inelastic scattering

33. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 33 Fission n absorbed composed nucleus n reemitted + deexcitation of the nucleus by fragmentation in 2 (or 3) lighter nuclei Possible threshold (no threshold  fissile nucleus) Profile of  f ~   : region in 1/v, then resonances less and less separated, but then limited variations

34. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 34 I.3 NUCLEAR FISSION DESCRIPTION Energetic feasibility Separation energy S n of a n (B(A,Z) : binding E of the nucleus (A,Z))  S n < average binding E per nucleon for a heavy nucleus (see. Graphic § I.1) But S n : minimum E to provide to a nucleus (A-1,Z) to form a composed nucleus (A,Z) < 0

35. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 35 Semi – empirical formula for the nucleus mass Heavy nuclei: 1 st approx. B(Z,A)  A  m(nucleus)  A Nucleus radius:R = r o.A 1/3  V(nucleus)  A  Density +/- cst  Nuclei ~ drops of incompressible liquid  Semi-empirical formula of Weizsäcker : Superficial tension Coulombian repulsion N – Z asymmetrySpin parity factor Dominant for heavy nuclei

36. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 36 Addition of a n to a nucleus: ‘brings’ S n + E kin a.Nucleus (Z even, N odd)  nucleus (Z even, N even)  + a p / A 3/4 b.Nucleus (Z even, N even)  nucleus (Z even, N odd)  - a p / A 3/4 Difference of S n = 2. a p / A 3/4 ~ O(MeV) ! In case b., about 1 MeV of addittional E kin necessary to excite the composed nucleus from (A,Z) Rem: case a. : U 235, U 233, Pu 239  fissile case b. : U 238, Th 232  fertile  fission threshold: 0.8MeV

37. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 37 Spontaneous vs. induced fission For (A,Z)  (A 1,Z 1 ) + (A 2,Z 2 ) We have m nucleus (A,Z) >  i m nuclei (A i,Z i ) (for large A, see curve B(A,Z)/A)  spontaneous fission? possible but not observed See potential energy of the fragments as a function of d E c = Coulomb potential in d = R 1 + R 2 between Z 1 and Z 2 charges E f = diff. binding E between (A,Z) and the fragments Nuclear forces Coulombian repulsion

38. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 38 Tunnel effect? Nuclei ‘a bit’ too heavy…  E brought by n (S n + E kin ) to overcome E c - E f (or E of a  ) Ex: symmetrical fission E f > E c for A > ~ 260  Unstable nuclei For A  [230,240] S n ~ 5 to 6 MeV E c – E f ~ 5.5 to 6 MeV  Induced fission possible Rem: what about1 st divergence of a reactor?

39. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 Development of an induced fission ((n,f) capture) Absorption time of a n:  10 -17 s Lifetime of an excited composed nucleus: ~ 10 -14 s Agitation of the nucleons in the excited nucleus  Formation of 2 fragments (with various (A i,Z i ))  Coulombian repulsion: ~ 10 -20 s Unstable fragments (N/Z ratio outside equilibrium)  Deexcitation by emission of prompt n (~ 10 -17 s) Spectrum of the emitted n (prompt fission spectrum): +/- Maxwell distribution 39 (  fission n indep. of the charact. of the absorbed n) (isotropic spectrum) (units?)

40. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 40

41. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 41 Mass distribution of the fission fragments

42. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 42 E excitation (fragments) : not fully consumed by n emission  Emission of  (called prompt, ~ 10 -14 s) E kin (fragments) :  Larger part of the E released by fission  Lost per ionization and excitation of the atoms in the media crossed  Fragments  fission products unstable because of lack of Z disintegrations  Remark: ~ 30 possible fission modes + disintegrations  mix of ~ 180 different radioactive nuclei!

43. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 43 DELAYED NEUTRONS Origin Fission products in radioactive chains in an excited state:  Usually  emission  Sometimes (i.e. if E excitation sufficient) emission of n  delayed neutrons and  Emission time? Linked to the half-lifetime of the previous isotope (precursor) in the chain (because deexcitation time much shorter)

44. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 44 Numerous precursors (> 50) but grouped in 6 classes characterized by oHalf-lifetime T i of the precursor oFraction  I of fission neutrons in group i oAverage E i with spectrum  i (E) Distribution of the fission energy Production delayed Residual heat to be evacuated even after reactor shutdown (  =  i  i = 0.68 %)

45. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 45 Evolution of the neutron population N : expected lifetime of a n / cycle ~ 10 -4 s   If k eff = 1.001, we have N(1 s) / N(0) ~ e +10 !!! But influence of the delayed n:  If k eff = 1.001, we have N(1 s) / N(0) ~ e +0.01 !!!  Delayed n compulsory for reactor control  If k eff = 1 / (1 -  ) : criticality reached without delayed n  prompt-critical threshold to avoid!!

46. PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016 46 CH.I : INTRODUCTION TO REACTOR PHYSICS FROM THE FISSION PROCESS TO THE REACTOR CHARACTERISTICS DEFINITION NUCLEAR FUELS n – heavy nucleus INTERACTION NEUTRON CYCLE AND CRITICALITY CONSTITUTIVE ELEMENTS OF REACTORS CROSS SECTIONS PROFILES INTERACTION MECHANISMS ASSOCIATED CROSS SECTIONS NUCLEAR FISSION DESCRIPTION DELAYED NEUTRONS   