Reactor Physics: Basic Definitions and Perspectives. Table of Contents

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1 Reactor Physics - Basic Definitions and Perspectives Reactor Physics: Basic Definitions and Perspectives prepared by Wm. J. Garland, Professor, Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada More about this document Summary: The basic definitions and perspectives for the behaviour of free neutrons as they interact with their surrounding media are introduced. This forms the basis for the detailed study to follow. Table of Contents Introduction Overview Learning outcomes To understand the following physical processes To understand the basics of neutron processes To understand the main issues for reactor modelling The Life and Times of the Neutron The Fission Event Neutron Life Cycle in CANDU Density of neutrons required to produce watt/cm Neutron Energy Units... 3 /E Spectrum Derivation of /E spectrum (equation 8-4 of D & H) Decay Math aside Example (D&H #2.3) Cross Section Microscopic cross section, σ [cm 2 ] Example (D & H 2.7) Macroscopic Cross Section, Σ [cm - ] Mean Free Path Calculation of Nuclei Density Nuclear Reactions Summary Summary of key concepts Summary of approximations A Look Ahead The neutron balance The Central Role of Flux... 28

2 Reactor Physics - Basic Definitions and Perspectives 2 9 Some Questions Question on characteristics Reactor Modelling Issues Question of n(e) Question of non-maxwellian Question on Cross section List of Figures Figure Course Overview... 3 Figure 2 The fission event... 5 Figure 3 Fission cross section of U-235 [source: DUD976, figure 2-7]... 5 Figure 4 Neutron life cycle [source: unknown]... 6 Figure 5 Another view of the neutron life cycle [source: EP72 course notes, chapter 2]... 7 Figure 6 Neutron Energy Distribution... 0 Figure 7 Nuclear Transformations [Source: A. A. Harms, McMaster University] Figure 8 Segment of the Chart of the Nuclides [Source: A. A. Harms, McMaster University] Figure 9 Number of neutrons and protons in stable nuclei [Source: A. A. Harms, McMaster University] Figure 0 Neutron processes Error! No table of figures entries found. List of Tables

3 Reactor Physics - Basic Definitions and Perspectives 3 Introduction. Overview You are here Reactor Physics basics statics kinetics dynamics systems basic definitions one speed diffusion pt kinetics feedback MNR nuclear interactions -d reactor depletion heat transfer CANDU neutron balance multigroup Poison thermal hydraulics chain reactions numerical criticality numerical methods cell calculations Figure Course Overview

4 Reactor Physics - Basic Definitions and Perspectives 4.2 Learning outcomes.2. To understand the following physical processes fission neutron life cycle the neutron environment neutron energy distribution.2.2 To understand the basics of neutron processes decay absorption and scattering kinematics.2.3 To understand the main issues for reactor modelling

5 Reactor Physics - Basic Definitions and Perspectives 5 2 The Life and Times of the Neutron 2. The Fission Event Fission ~200 MeV in kinetic energy The neutron, which is uncharged, can interact with a U 235 nucleus leading to fission. thermal neutron U 235 nucleus Fission Products 2-3 fast neutrons The result is the creation of fission products (which may be radioactive), radiation (usually γ s and β s) and 2 to 3 neutrons at high energy (-2 MeV). Before After The probability of the event is a strong function of the neutron energy, as shown next. Figure 2 The fission event Figure 3 Fission cross section of U-235 [source: DUD976, figure 2-7] σ microscopic cross section [cm 2 ] = effective interaction area barn 0 cm σ is usually quoted in units of barns since the effective area is so small.

6 Reactor Physics - Basic Definitions and Perspectives Neutron Life Cycle in CANDU Figure 4 Neutron life cycle [source: unknown]

7 Reactor Physics - Basic Definitions and Perspectives 7 When Number of Slow Neutrons is Leaked Neutrons constant, the system is critical. Delayed Neutrons appear after Neutrons Slowing ~ 0 seconds. Down Fast Neutrons slow down in ~ millisecond. Delayed Neutrons from Fission Prompt Neutrons from Fission "ASHES" (Fission Products) HEAT U235 Fission Neutrons Diffusing Leaked Neutrons CONTROL THIS TO CONTROL HEAT PRODUCTION Captured Slow Neutrons Neutrons (Some neutrons are captured in U238 and so produce useful fuel - Pu239) Figure The Neutron Cycle in a Thermal Reactor Figure 5 Another view of the neutron life cycle [source: EP72 course notes, chapter 2]

8 Reactor Physics - Basic Definitions and Perspectives Density of neutrons required to produce watt/cm 3 Consider a beam of neutrons moving at velocity, v cm/s. The average distance travelled before a fission interaction is X cm (in U 235 ) v Average time per interaction = X / v seconds and frequency of interaction = s - per neutron. x If the density of neutrons is n neutrons/cm 3, then interaction rate = nv x interactions/s-cm3. For watt 3 cm Joule = s cm 3 J energy = s cm 3 fission, #fissions s cm Joules n v = ev ev x n = x ~ cm v ~2 0 5 cm/s n/cm Compare this to the typical nuclei density 22 3 ~ 0 /cm Conclusion: Neutrons do not interact with each other. This is an important conclusion.

9 Reactor Physics - Basic Definitions and Perspectives Neutron Energy Thermal distribution: 3/2 m 2 n(v) = 4π n 0 v e 2π kt 2πn (π kt) 0 /2 n(e) = E e 3/2 0 -E/kT φ(e) n(e) v = vn M(E) /2 2πn0 2 = E e 3/2 (πkt) m -mv 2 /2kT -E/kT E = mv 2 2 Maxwellian Distribution Now, n = n(e)de = n(v)dv 0 o Note: n(e) = # of neutrons in interval de [# / ev] o n(v) = # of neutrons in interval dv [# / (m/s)] Thus 2 2 n mv n(v ) since interval size is different But n(e) d(e) = n(v)dv so that n(e)de= 0 0 n(v)dv Most probable vel: dn(v) = 0 v = dv p 2kT m Most probable energy: dn(e) = 0 E = kt d E p 2 = 2200 m/s E = 3 2 kt E(v ) = kt = 0.025eV at 20 o C p v = 8kT πm

10 Reactor Physics - Basic Definitions and Perspectives 0 Figure 6 Neutron Energy Distribution

11 Reactor Physics - Basic Definitions and Perspectives 2.5 Units v 23 2kT 2 x.3806x0 Joules / K x 293.3K p = = 27 m.67x0 kg = 220 m / s 2 2 cm m E [ ] gm = erg or kg = Joules 2 2 sec sec Recall: F = ma dyne = gm cm/sec -7 E = F i x = dyne - cm = erg = 0 J. 2

12 Reactor Physics - Basic Definitions and Perspectives 2 3 /E Spectrum 3. Derivation of /E spectrum (equation 8-4 of D & H) Assume the neutron is slowing down in H in the absence of absorption. Further assume that there is no upscatter. [Σ s (E) + Σ a (E)] φ (E) = Σ (E E) φ (E ) de + S(E) E s # of neutrons leaving energy E # scattering down to energy E Since a (E) = 0, we have Σ (E ) Σ s (E) φ (E) = s φ (E )de + S(E) E E F(E) equal probability of scatter (isotropic) E 0 F(E ) F(E) = de + S 0 δ (E - E 0) E E d F(E) = F(E) d E E φ S E 0 F(E) = + S 0 δ (E - E 0) S 0 0 (E) = + δ(e-e 0) Σ s(e) E Σ s(e) S At this point you should be able to answer Questions, 2, 3 and 4 at the end of this chapter. Σ ~ const φ ~ s E

13 Reactor Physics - Basic Definitions and Perspectives 3 4 Decay - d N(t) = λ N (t) dt N(t) = N e 0 -λt dn(t) dt = RATE = λ N e -λt 0 # decaying in dt at t = - dn (t) 0 -λt = λ N e dt -λt fraction of initial decaying in dt at t = λ e dt = p(t) dt probability mean lifetime, t = p(t) tdt = λ te dt = 0 0 λ -λt t = λ Half Life, T /2 N0 N(T /2) = = N 0 e 2 -λt /2 ln T /2 = = λ λ

14 Reactor Physics - Basic Definitions and Perspectives 4 4. Math aside If we have two functions f(x) + g(x): d(fg) = f g + g f d(fg) = fg = f gdx + g fdx -λt te e 0 t e dt = dt λ 0 f g' (-λ) 0 = 0 + λ 2 -λt -λt

15 Reactor Physics - Basic Definitions and Perspectives Example (D&H #2.3) Decay chain for an initially pure radioactive sample. dn = dt λ N -λ t N (t) = N (0) e dn 2 = λ N λ N dt 2 2 λ N (0) -λ t -λ t N (t) = [e e 2 ] 2 λ - λ 2 dn 3 = λ N - λ N dt dn N = λ N - λ N dt N- N- N N messy solutions To solve for N 2 : Rewrite to get: dn 2 + λ N = λ N dt 2 2 λ Multiply by e 2t : d N + λ N dt = λ N dt λ t λ t λ t e 2 dn + λ e 2 N dt = λ N e 2 dt λ t (λ - λ )t d(e 2 N ) = λ N (0)e 2 dt 2 λ t λ N (0) (λ - λ )t e 2 N = e λ - λ C 2

16 Reactor Physics - Basic Definitions and Perspectives 6 Now at -λ N (0) t = 0, N (0) = 0 C = 2 λ - λ 2 λ N (0) (λ - λ )t -λ t λ N (0) -λ t -λ t N (t) = [e 2 -] e 2 = [e - e 2 ] 2 λ - λ λ - λ 2 2 For fast decay of and slow decay of 2 ( λ >> λ ) 2 N(t) ~ 2 λ N (0) -λt -λ2t [ e - e ] λ - λ λ t 2 ~ N (0) e i.e. decay dominated by decay of 2. For slow decay of and fast decay of 2 ( λ >> λ ) 2 λ N (0) -λ t λ N (t) ~ e = N (t) 2 λ λ 2 2 i.e. λ N (t) = λ N(t) 2 2 This is called secular equilibrium. (lasting a long time, indifferent, not religious)

17 Reactor Physics - Basic Definitions and Perspectives 7 5 Cross Section 5. Microscopic cross section, σ [cm 2 ] Consider a beam of neutrons incident on a target. The rate of interaction (neutron-nuclei) is # Rate of interaction =σ I N [ ] cm 3 s Recall that barn = 0-24 cm 2 2 cm # # 2 cm s 3 cm The total cross section, σ = σ + σ total scatter absorption ie. σ = σ + σ T s a Total scatter absorption inelastic elastic fission capture (n, γ) (n, 2n) (n, 3n) (n, p) (n, α)

18 Reactor Physics - Basic Definitions and Perspectives Example (D & H 2.7) Question: How long, on average for a given nuclei to suffer a neutron interaction? Rate = σi N = interactions/sec = interactions/sec for nuclei seconds/interactions for nuclei = = seconds s

19 Reactor Physics - Basic Definitions and Perspectives Macroscopic Cross Section, Σ [cm - ] Rate = σ I N Σ I = di dx cm 2 i # Σ σ N cm 3 - = cm di ( ) I = Σ dx = fractional change of I in distance dx = probability of reaction per unit length -Σx I(x) = I e o I(x) I o = probability of going x with no interaction -Σx = e Probability of interaction at x in dx is: (p(x)dx) At this point you should be able to answer Questions 5 at the end of this chapter. di I(x) - = Σdx = Σe -Σx d I I o o x p(x)

20 Reactor Physics - Basic Definitions and Perspectives Mean Free Path x = p(x) xdx = Σ e -Σx xdx 0 0 = = Σ mean free path -λt cf: t = λe tdt = 0 λ Mean time between collisions: x = velocity vσ Collision frequency = time = vσ 5.5 Calculation of Nuclei Density Σ = Σ + Σ Σ = N σ i t a s s i s i y Σ = N σ x + N σ +... a x a y a = N σ i i a i # gm A ρ gm - mole i cm 3 N =, where A = Avogadro's number, x0 i gm A gm - mole 23

21 Reactor Physics - Basic Definitions and Perspectives 2 6 Nuclear Reactions Reactions: a+b a (b,c) d c+d c Example: n + U 235 U γ b a d U (n, γ) U Radioactive Capture: (n, γ) Before After Fission: n + X X + X + ν n + energy A X Z Scattering: (n,n) elastic (n, n ) inelastic Source of neutrons:. Fission A. Initiated by cosmic radiation B. Spontaneous C. Neutron absorption X = some nucleus Z = Atomic number = number of protons N = number of neutrons A = Mass number = N+Z = total number of nucleons 2. (α,n) Be 9 + He 4 C 2 + n from radium, Pu, polonium 3. (γ,n) (photoneutrons) H 2 + γ H + n 0 Be 9 + γ Be 8 + n 4 4 0

22 Reactor Physics - Basic Definitions and Perspectives 22 Figure 7 Nuclear Transformations [Source: A. A. Harms, McMaster University]

23 Reactor Physics - Basic Definitions and Perspectives 23 Figure 8 Segment of the Chart of the Nuclides [Source: A. A. Harms, McMaster University]

24 Reactor Physics - Basic Definitions and Perspectives 24 Figure 9 Number of neutrons and protons in stable nuclei [Source: A. A. Harms, McMaster University]

25 Reactor Physics - Basic Definitions and Perspectives 25 7 Summary 7. Summary of key concepts - neutrons do not interact with each other - life cycle - neutron energy spectrum - decay - chart of the nuclides - σ, - mechanics of collision (addendum)

26 Reactor Physics - Basic Definitions and Perspectives Summary of approximations

27 Reactor Physics - Basic Definitions and Perspectives 27 8 A Look Ahead 8. The neutron balance Sources and Absorption Diffusion Transport Figure 0 Neutron processes Neutron balance: n ( r, t) φ( r, t) = S( r, t) Σ φ ( r, t) - i J ( r, t) (2) t v t a =+ id φ( r,t) from Fick's Law : J -D φ Reactor physics is all about the calculation of the neutron density, n, or flux, φ.

28 Reactor Physics - Basic Definitions and Perspectives The Central Role of Flux Σ φ S vσ f Fick s Law sources v φ t Diffusion Neutron balance eqn. (continuity eqn.) # Dim. SS or tran. Delayed precursors # of groups variance in material properties. Sinks, Σ a, Σ s -depletion -inhomo -control -power dependence (Temperature effects)

29 Reactor Physics - Basic Definitions and Perspectives 29 Mat. Lib. - Say 69 groups Fuel Control Moderator Cell Materials & Composition t/h input Effect on samples (γ, heating, φ, etc.) Cell Defn (group collapse) Grid Dimensions # of grid pts. Type of cell at each grid pt. t/h calc. T,P, density flows Radiation damage on structure Grid Prop Setup Initial Flux Burnup Fuel Management Flux calc s - Multigroup diffusion - Transport - Monte Carlo φ g (r, t, E, Ω) N i (r, t) C i (r, t) Fission Products - Xenon - delayed precursors Detector Response Control Rod position

30 Reactor Physics - Basic Definitions and Perspectives 30 9 Some Questions 9. Question on characteristics Given this brief look at neutrons and their life cycle, what are some of the issues/characteristics that you would expect to arise in the design of a nuclear power plant? 9.2 Reactor Modelling Issues Imagine a reactor consisting of a central fuel region surrounded by a moderator. There is a variable absorber for control. What are some of the issues to consider in setting up a model of the reactor? 9.3 Question of n(e) Illustrate on a graph of n(e) vs. ln(e) the life cycle of a neutron in a fission reactor. 9.4 Question of non-maxwellian Illustrate how the thermal neutron spectrum differs from a Maxwellian and explain why. 9.5 Question on Cross section Consider: I(x) = I e -Σx o What are some of the assumptions in or limitations of this equation? What are some of the things implied by this equation?

31 Reactor Physics - Basic Definitions and Perspectives 3 About this document: Author and affiliation: Wm. J. Garland, Professor, Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada back to page Revision history: Revision.0, September 4, 2004, initial creation from hand written notes. Source document archive location: D:\TEACH\EP4D3\text\2-basic\basic-r.doc Contact person: Wm. J. Garland Notes: