Nuclear Fission. 1/v Fast neutrons. U thermal cross sections σ fission 584 b. σ scattering 9 b. σ radiative capture 97 b.

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1 Nuclear Fission 1/v Fast neutrons should be moderated. 235 U thermal cross sections σ fission 584 b. σ scattering 9 b. σ radiative capture 97 b. Fission Barriers 1

2 Nuclear Fission Q for 235 U + n 236 U is MeV. Table 13.1 in Krane: Activation energy E A for 236 U 6.2 MeV (Liquid drop + shell) 235 U can be fissioned with zero-energy energy neutrons. Q for 238 U + n 239 U is 4.??? MeV. E A for 239 U 6.6 MeV MeV neutrons are needed. Pairing term: δ =??? (Fig in Krane). What about 232 Pa and 231 Pa? (odd Z). Odd-N nuclei have in general much larger thermal neutron cross sections than even-n nuclei (Table 13.1 in Krane). Nuclear Reactors, BAU, 1 st Semester,

3 Nuclear Fission Why not use it? σ f,th x b Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 3

4 Nuclear Fission 235 U + n 93 Rb Cs + 2n Q =???? What if other fragments? Different number of neutrons. Take 200 MeV as an average. 66 MeV 98 MeV Heavy fragments Light fragments miscalibrated 4

5 Mean neutron energy 2 MeV. 2.4 neutrons per fission (average) 5 MeV average kinetic energy carried by prompt neutrons per fission. HW 45 Nuclear Fission Show that the average momentum carried by a neutron is only 1.5 % that carried by a fragment. Thus neglecting neutron momenta, show that the ratio between kinetic energies of the two fragments is the inverse of the ratio of their masses. E1 m E 2 m 1

6 Nuclear Fission Enge Distribution of fission energy Krane sums them up as β decays. Lost! 6

7 Nuclear Fission Segrè Distribution of fission energy Lost! a b c How much is recoverable? What about capture gammas? (produced by Whyc<(a+b a+b) )? (produced by ν-1 1 neutrons) Nuclear Reactors, BAU, 1 st Semester,

8 Nuclear Fission Recoverable energy release 200 MeV per 235 U fission. Fission rate = 2.7x10 21 P fissions per day. P in MW. Burnup rate: 1.05 P g/day. P in MW. Capture-to-fission ratio: σ γ ( E) α( E) = HW 46 σ (E) E Consumption rate: 1.05(1+α) P g/day MW reactor. 3.1x10 19 fissions per second, or gram of 235 U per second. Two neutrinos are expected immediately from the decay of the two fission products, what is the minimum flux of neutrinos expected at 1 km from the reactor. Nuclear Reactors, BAU, 1 st Semester, σ f 4.8x10 12 m -2 s -1 8

9 Nuclear Fission 3.1x10 10 fissions per second per W. In thermal reactor, majority of fissions occur in thermal energy region, φ and Σ are maximum. Total fission rate in a thermal reactor of volume V V Σ f φ Thermal reactor power (quick calculation) P th = V Σ f φ 3.1 x Nuclear Reactors, BAU, 1 st Semester,

10 Controlled Fission Fast second generation neutrons 235 U + n X + Y + (~2.4)n Moderation of second generation neutrons Chain reaction. Net change in number of neutrons from one generation to the next k (neutron reproduction factor). k 1 Chain reaction. Water, D 2 O or graphite moderator. k < 1 subcritical system. k = 1 critical system. k >1 supercritical system. For steady release of energy (steadystate operation) we need k =1. Infinite it medium (ignoring i leakage at the surface). Chain reacting pile 10

11 Controlled Fission 235 U thermal cross sections σ fission 584 b. σ scattering 9 b. σ radiative capture 97 b. Probability for a thermal neutron to cause fission on 235 U is σ f 1 = σ + σ α γ 1+ f γ + If each fission produces an average of ν neutrons, then the mean number of fission neutrons produced per thermal neutron = η σ η = ν σ σ ν = 1 α f f = ν η <ν a σ f + σ γ + Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 11

12 Controlled Fission Assume natural uranium: % 238 U, % 235 U. Thermal σ f = 0 b Thermal σ γ = 2.75 b Σ = Σ x x x + Σ y y = N y x σ = ( γ σ + γ σ ) N x + N Σ f / N = ( )(0) + (0.0072)(584) ) = 4.20 b. Σ γ / N = ( )(2.75) + (0.0072)(97) 0072)(97) = 3.43 b. 584 b 97 b y σ Nuclear Reactors, BAU, 1 st Semester, y 2 4πR 4 π R 2 Using the experimental elastic scattering data the radius of the nucleus can be estimated. 235 U 238 U Doppler effect? 12

13 Moderation (to compare x-section) (n,n) n) 2 H (n,n) 1 H (n,γ) (n,γ) Resonances?

14 Controlled Fission Probability for a thermal neutron to cause fission For natural uranium = σ f σ + +σ 4.20 f γ = If each fission i produces an average of ν = 2.4 neutrons, then the mean number of fission neutrons produced per thermal neutron = η = 2.4 x σ f η = =ν σ f + This is close to 1. If neutrons are still to be lost, there is a danger of losing criticality. For enriched uranium ( 235 U = 3%) η =????? (> 1.3). In this case η is further from 1 and allowing for more neutrons to be lost while maintaining criticality. σ γ 14

15 Controlled Fission N thermal neutrons in one generation have produced so far ηnη fast neutrons. Some of these fast neutrons can cause 238 U fission more fast neutrons fast fission factor = ε (= 1.03 for natural uranium). Now we have εηn fast neutrons. We need to moderate these fast neutrons use graphite for 2 MeV neutrons we need??? collisions. How many for 1 MeV neutrons? The neutron will pass through the ev region during the moderation process. This energy region has many strong 238 U capture resonances (up to 1000 b) Can not mix uranium and graphite as powders. In graphite, an average distance of 19 cm is needed for thermalization the resonance escape probability p ( 0.9). 15

16 Controlled Fission Now we have pεηn thermal neutrons. Graphite must not be too large to capture thermal neutrons; when thermalized, neutrons should have reached the fuel. Graphite thermal cross section = b, but there is a lot of it present. Capture can also occur in the material encapsulating the fuel elements. The thermal utilization factor f ( 0.9) gives the fraction of thermal neutrons that are actually available for the fuel. Now we have fpεηn thermal neutrons, could be > or < N thus determining i the criticality of the reactor. The four-factor factor formula. k = fpεη k = fpεη(1-l fast )(1-ll thermal ) Fractions lost at surface 16

17 Neutron reproduction factor k = x 0.9 Thermal utilization i factor f ili i x η What is: Migration length? Critical size? How does the geometry affect the reproduction factor? x 0.9 Resonance escape probability p x 1.03 Fast fission factor ε 17

18 Controlled Fission Time scale for neutron multiplication Time τ includes moderation time (~10-6 s) and diffusion time of thermal neutrons (~10-3 s). Time Average number of thermal neutrons t N t + τ kn t + 2τ k 2 N For a short time dt Show that N ( t ) dn dt = = kn N τ N 0 e ( k 1) t τ 18

19 Controlled Fission N ( t ) = k = 1 N is constant (Desired). N 0 e ( k 1) t k < 1 N decays exponentially. k >1 N grows exponentially with time constant τ /(k (k-1). k = 1.01 (slightly supercritical) e (0.01/0.001)t = e 10 = in 1s. Cd is highly absorptive of thermal neutrons. Design the reactor to be slightly subcritical for prompt neutrons. The few delayed neutrons will be used to achieve criticality, allowing enough time to manipulate the control rods. Cd control rods τ 19

20 Fission Reactors Essential elements: Fuel (fissile material). Core Moderator (not in reactors using fast neutrons). Reflector (to reduce leakage and critical size). Containment vessel (to prevent leakage of waste). Shielding (for neutrons and γ s). Coolant. Control system. Emergency systems (to prevent runaway during failure). 20

21 Fission Reactors Types of reactors: Used for what? Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity. Research reactors: low power (1-10 MW) to generate neutrons (~10 13 n.cm -2.s -1 or higher) for research. Converters: Convert non-thermally-fissionable i material to a thermally-fissionable material. 238 U + Fertile n 239 U 23min 239 Np + β 2.3d _ + ν 239 Pu σ f,th = 742 b _ + β + +ν 21

22 Fission Reactors 232 Th + n Fertile 233 Th 22 min 233 Pa _ + β +ν 27d 233 U + β _ + ν σ f,th = 530 b If η = 2 Conversion and fission. If η > 2 Breeder reactor. 239 Pu: Thermal neutrons (η = 2.1) hard for breeding. Fast neutrons (η = 3) possible breeding fast breeder reactors. After sufficient time of breeding, fissile material can be easily (chemically) separated from fertile material. Compare to separating 235 U from 238 U. 22

23 Fission Reactors What neutron energy? Thermal, intermediate (ev kev), fast reactors. Large, smaller, smaller but more fuel. What fuel? Natural uranium, enriched uranium, 233 U, 239 Pu. How??? From converter or breeder reactor. HW 47 23

24 Fission Reactors What moderator? 1. Cheap and abundant. 2. Chemically stable. 3. Very low mass (~1). 4. High density. 5. Minimal neutron capture cross section. Graphite(1245)increaseamount (1,2,4,5) increase amount to compensate 3. Water (1,2,3,4) but n + p d + γ enriched uranium. D 2 O (heavy water) has low capture cross section natural uranium, but if capture occurs, produces tritium. Be and BeO, but poisonous. 24

25 Fission Reactors What assembly? Heterogeneous: moderator and fuel are lumped. Homogeneous: moderator and fuel are mixed together. In homogeneous systems, it is easier to calculate p and f for example, but a homogeneous natural uraniumgraphite mixture can not go critical. What coolant? Coolant prevents meltdown of the core. It transfers heat in power reactors. Why pressurized-water reactors. Why liquid sodium? 25

26 Boiling water reactor Pressurized ri ed water reactor Light water reactors. Both use light water as coolant and as moderator, thus enriched (2-3%) uranium is used. Common in the US. 26

27 CANDU reactor Canada has D 2 O and natural uranium. Most power reactors in GB are graphite moderated gas- cooled. Gas cooled reactor 27

28 Liquid sodium cooled, fast breeder reactor. Blanket contains the fertile 238 U. Water should not mix with sodium. Breeder reactor 28