for small r

1. While for large r, after the fragments have been scissioned  r r r for small r V(r) for large r separation r

2. For such quadrupole distortions the figure shows the energy of deformation (as a factor of the original sphere’s surface energy Es) plotted against  for different values of the fission parameter x. When x > 1 (Z2/A>49) the nuclei are completely unstable to such distortions.

3. Z2/A=36 such unstable states decay in characteristic nuclear times ~10-22sec Z2/A=49 Tunneling does allow spontaneous fission, but it must compete with other decay mechanisms (-decay) The potential energy V(r) = constant-B as a function of the separation, r, between fragments.

5. Thermal neutrons E< 1 eV Slow neutrons E ~ 1 keV Fast neutrons E ~ 100 keV – 10 MeV Typical of decay Products & nuclear reactions The incident neutron itself need not be of high energy. “Thermal neutrons” (slowed by interactions with any material they pass through) have been demonstrated to be particularly effective. Cross section incident particle velocity, v This merely reflects the general ~1/v behavior we have noted for all cross sections!

6. At such low excitation there may be barely enough available energy to drive the two fragments of the nucleus apart. Division can only proceed if as much binding energy as possible is transformed into the kinetic energy separating them out. (so MOST of the available Q goes into the kinetic energy of the fragments!) Thus the individual nucleons settle into the lowest possible energy configurations involving the most tightly bound final states.

7. There is a strong tendency to produce a heavy fragment of A ~ 140 (with double magic numbers N = 82 and Z = 50).

8. Recall A possible (and observed) spontaneous fission reaction 8.5 MeV/A 7.5 MeV/A Gains ~1 MeVper nucleon! 2119 MeV = 238 MeV released by splitting 238U 119Pd

9. 238 MeV represented an estimate of the maximum available energy for symmetric fission. For the observed distribution of final states the typical average is ~200 MeVper fission. This 200 MeV is distributed approximately as:

10. 235U

11. Recall Isobars off the valley of stability (dark squares on preceding slide) b-decay to a more stable state.

12. Recall a and b decays can leave a daughter in an excited nuclear state 1/2- 2- b- b- 187W 198Au 0.68610 1.088 MeV 0.61890 b- b- 0.20625 0.412 MeV 0.13425 5/2+ 0+ 187Re 198Hg

13. With the fission fragments radioactive, a decay sequence to stable nuclei must follow

14. With the fission fragments radioactive, a decay sequence to stable nuclei must follow 25 sec b,g 18 min b,g 4 hr b,g 33 d b,g 0.03% 65 sec b,g 13 d b,g 40 h b,g 6 sec b,g 7 min b,g 10 hr b,g 106 yr b,g 1.40% 5 sec b,g 3 hr b,g 4 h b,g sometimes or

15. For 235U fission, average number of prompt neutrons ~ 2.5 with a small number of additional delayedneutrons.

16. with every neutron freed comes the possibility of additional fission events This avalanche is the chain reaction.

17. 235U will fission (n,f) at all energies of the absorbed neutron. It is a FISSILE material. However such a reaction cannot occur in natural uranium (0.7% 235U, 99.3% 238U)

18. Total (t) and fission (f) cross sections of 235U. 1 b = 10-24 cm2

20. Notice: 238U has a threshold for fission (n,f) at a neutron energy of 1MeV. The difference between these two isotopes of uranium is explained by the presence of the pairing term in the semi-empirical mass formula. for Z even, N even for Z odd, N odd for A odd Like nucleons couple pairwise into especially stable configurations.

21. Note the strong resonant capture of neutrons (n, ) in the energy range 10-100 eV (particularly for 238U where the cross-section reaches high values)

22. The fission neutron energy spectrum peaks at around1 MeV

23. At 1 MeV the inelastic cross-section (n,n') in 238U exceeds the fission cross-section. This effectively prevents fission from occurring in 238U.

24. only the Natural uranium (0.7% 235U, 99.3% 238U) undergoes thermal fission Fission produces mostly fast neutrons Mev but is most efficiently induced by slow neutrons E (eV)

25. Consider fission neutrons created deep enough in a lump of natural uranium that we’ll just (for now) ignore that some neutrons may simply escaping from the sample.

26. 1000 The processes competing with neutron-induced fusion have approximate cross-sections (read from the graphs at right) of 100 Cross-section (barns) 10 1 235U (n,n) elastic scattering~ 5 barn (n,n’) inelastic scattering~ 3 barn (n,) ~0.2barn (n,f) fission ~ 2 barn 10000 1000 100 Cross-section (barns) 238U (n,n) elastic scattering~ 5 barn (n,n’) inelastic scattering~ 2 barn (n,) ~0.2barn (n,f) fission ~0.6barn 10 1 0.1

27. Giving a relative probability to each of: 235U (n,n) elastic scattering 6.7 (n,n’) inelastic scattering1.7 (n,) 0.3 (n,f) fission 3.3  0.7/99.3 238U (n,n) elastic scattering8.3 (n,n’) inelastic scattering3.3 (n,) 0.3 (n,f) fission 1

28. Of the first 100 fission neutrons we start with ~98 are captured in the dominant 238U 238U (n,n) elastic scattering63 (n,n’) inelastic scattering 25 (n,) 2 (n,f) fission 8 only 8 of these captures result in fission 235U (n,n) elastic scattering 1 (n,n’) inelastic scattering0 (n,) 0 (n,f) fission 0 With2-3 neutrons generated by each fission, only ~20 neutrons in the second generation - this is insufficient to sustain a chain reaction.

29. Enriching the 235U content FAST REACTOR a 50-50 mix of the two isotopes will sustain a chain reaction (most fission events occurring now in 235U by neutron energies in the range 0.3 - 2.0 keV. THERMAL REACTOR moderating the neutrons to thermal speeds mixing natural uranium with a material to slow (but not absorb) neutrons to lower energies where the fission cross-section for 235U is large. Most fissions are then induced by neutrons with thermal energies (~0.025 eV).

30. Powdered uranium Granulated powders can be mixed for this purpose. Or blocks of uranium fuel can be alternately stacked with graphite to form a nuclear pile.

31. 1. Starting with  neutrons/fission FUEL 2. Avg of  neutrons after fast fission 3. p survive thermalization Moderator (Graphite) 4. pf number captured in 235U FUEL 238U Moderator (Graphite) FUEL 5. k =pf(f /total) number producing fission 235U

32. One fission event produces k =pf(f /total) secondary fission events. k is the reproduction factor. A chain reaction requires k1. If k=1 the core is “critical” and self-sustaining. Typical values for natural uranium/graphite piles are k=1.07

33. Uranium is not dumped into the core like coal shoveled into a furnace.  Instead it is processed and formed into fuel pellets (~pencil eraser size).  The fuel pellets are stacked inside hollow metal tubes to form fuel rods 11 to 25 feet in length.  Before it is used in the reactor, the uranium fuel is not very radioactive.

34. The fuelrods are arranged in a regular lattice inside the moderator. The rods are typically 2-3 cm in diameter and spaced about 25 cm apart. The rods metal sheath or cladding – most commonly stainless steel or alloys of zirconium. This cladding supports the fuel mechanically, prevents release of radioactive fission products into the coolant stream and provides extended surface contact with the coolant in order to promote effective heat transfer.

35. A single fuel rod cannot generate enough heat to make the amount of electricity needed from a power plant.  Fuel rods are carefully bound together in assemblies, each of which can contain over 200 fuel rods.  The assemblies hold the fuel rods apart so that when they are submerged in the reactor core, water can flow between them. In nuclear power plants, the moderator is often water (though some types do still use graphite).

36. Fuel cell channels in face of reactor core.

38. Control rods slide in or out between the fuel rods to regulate the chain reaction. contain cadmium or boron (high cross section for neutron absorption, without fission).  e.g., natural Boron is 20% 10B Control rods act like sponges to absorb excess neutrons. with a cross section for thermal neutrons of 3840 b for the process

39. When the core temperature drops too low, the control rods are slowly pulled out of the core, and fewer neutrons are absorbed.  When the temperature in the core rises, the rods are slowly inserted. To maintain a controlled nuclear chain reaction, the control rods are manipulated until each fission results in just one neutron on average, all other neutrons effectively absorbed by the control rods.Temperature changes in the core are generally very gradual.  However should monitors detect a sudden change in temperature, the reactor immediately shuts down automatically by dropping all the control rods into the core.  A shutdown takes only seconds and halts the nuclear chain reaction.

40. This very common type makes use of the excellent properties of water as both coolant and moderator (ordinary water does absorb neutrons – converting hydrogen into deuterium). The Boiling Water Reactor (BWR) allows the water to boil in the reactor core and uses the steam to drive the turbines.

41. The highest temperature possible for liquid water (critical temperature 374°C) is a limitation for devices that use water to convey heat. The core must be contained within a pressure vessel of welded steel (typically withstanding pressures of about 1.55 107 Pa or 153 bar. Furthermore recall: the Carnot engine efficiency   is In this ideal case the heat is received isothermally(the working fluid at T1) but rejected isothermally (at T2) with all processes reversible. No real power plant operates on an ideal Carnot cycle, but the expression shows the higher T1, the higher the efficiency (T2 cannot be lower than the outside temperature). 1st land based pressurized water reactor: Shippingport USA (1957).

42. Pressure vessels are enormous with 9 inch thick walls, often weighing more than 300 tons.  The pressure vessel surrounds and protects the reactor core, providing a safety barrier and holding the fuel assemblies, control rods, and coolant. Pressure vessels are made of carbon steel and lined with a layer of stainless steel to prevent rust.  The pressure vessel is located inside the containment building, a thick concrete structure reinforced with steel bars.

43. A Fast Reactor has no moderator and consequently a much smaller core.

44. The very high power involved means that liquid metals have to be used as coolants! Liquid sodium is the most common but has the disadvantage of becoming radioactive itself through 23Na(n, )24Na. As well as generating power fast reactors are used for breeding fissile material.

45. If uranium fission reactors used as sole source of electrical power needs all high-grade ores used up within a few decades! Breeder Reactors Fermi, Zinn (1944) Can fissile nuclei be grown? (the result of any nuclear reaction) Can we create fissile material as a by product of any reaction? The parent nuclei that spawns the fissile material is described as being a fertile nuclide.

46. Example: build a reactor core that runs on 239Pu (the fuel) packed within a bed of 238U (the fertile nuclide) • =2.91 fast neutrons/239PU fission Only one of these on average producing an additional fission is sufficient for sustainability. If the rest are incident on 238U there’s a chance of inducing 1/2= 25 min - 1/2= 2.3 days - A well designed breeder reactor can double the amount of fissile material in 7 – 10 years.