Lewis 2.1, 2.2 and 2.3

Transcription

1 Chapter 2(and 3)

2 Cross-Sections TA Lewis 2.1, 2.2 and 2.3

3 Learning Objectives Understand different types of nuclear reactions Understand cross section behavior for different reactions Understand d resonance behavior and its relation to the nuclear energy levels Know where to find nuclear data

4 Microscopic Cross section Consider a beam of mono-energetic neutrons of intensity I incident on a very thin material such that there are Na atoms/cm. 2.s The collision rate of neutrons is proportional to the neutron beam intensity and the nuclei density Na. The constant of proportionality is defined as the neutron microscopic cross-section." N A nuclei/cm 2 I neutrons/cm 2. sec Image by MIT OpenCourseWare. x Image by MIT OpenCourseWare.

5 Microscopic Cross section The microscopic cross-section characterizes the probability of a neutron interaction. R = σ I N [ A # ] [ ] [ # ] [ # ] cm2 cm2s cm2s cm2 N A nuclei/cm 2 I neutrons/cm 2. sec x Image by MIT OpenCourseWare.

6 Cross Section - Microscopic Scattering Absorption Total σ s = σ e + σ in σ a = σ γ + σ f σ t = σ s + σ a Number of reactions /nucleus / s (R / N A ) σ = = Number of incident neutrons / cm2s 1

7 Incident Beam (Neutron) on a Thick Target I 0 I(x) x = 0 x x + dx x Image by MIT OpenCourseWare. Now consider the case of a thick target with an incident beam I 0 for which we want to know the unattenuated beam intensity as a function of position I(x).

8 Unattentuated beam in target Taking an infinitesimally thin portion of the target, dx, allows us to use the previous analysis on dx between x and x + dx. I(x) N A nuclei/cm 2 x = 0 x x + dx x x Images by MIT OpenCourseWare.

9 Unattentuated beam in target I(x) N A nuclei/cm 2 x x = 0 x x + dx x Images by MIT OpenCourseWare. Number of target nuclei per cm 2 in dx is dn A = N dx where N = number density of the target nuclei in units cm-3.

10 Relating reaction rate to beam intensity The total reaction rate in dx can be defined as I(x) dr = σ t IdN A = σ t INdx I 0 x = 0 x x x + dx Image by MIT OpenCourseWare. Each neutron that reacts decreases the unattenuated beam intensity, thus -di(x) = - [I(x + dx) - I(x)] = σ t INdx

11 Macroscopic cross-section we can then solve this differential equation to get I(x) di(x) = - NσtI(x) dx we can then define the macroscopic cross-section such that I(x) = I 0 e -Nσ x t

12 Macroscopic cross section interpretation Σ t Probability per unit path length that the neutron will interact with a nucleus in the target. exp(-σ t x) Probability that a neutron will travel a distance x without making a collision. Σ t exp(-σ t x)dx Probability that a neutron will make its first collision in dx after traveling a distance x.

13 Mean free path of neutron x dx x p(x) = Σ t dx x exp(-σ t x) = 0 0 Σ t Interaction probability calculates the average distance a neutron travels before interacting with a nucleus 1 x = Σ t 1 Average distance traveled by a neutron before making a collision

14 Two fundamental aspects of neutron cross sections Kinematics of two-particle collisions Conservation of momentum Conservation of energy Dynamics of nuclear reactions Potential scattering Compound nucleus formation

15 Hydrogen x.s Cross Section (b) Energy (MeV) Image by MIT OpenCourseWare.

16 Potential scattering Before After Image by MIT OpenCourseWare. Hard sphere collision where the neutron bounces off the nucleus. The interaction time is approximately 10-17s.

17 Compound nucleus formation γ Before Compound Nucleus After A+1 ( ) Z X ( ) A Z X 1 A+1 n 0 Z X * Radiative capture Image by MIT OpenCourseWare. Neutron penetrates the nucleus and forms a compound nucleus (excited state). The compound nucleus regains stability by decaying. The interaction time is approximately 10-14s.

18 Compound nucleus decay processes 1 n + A X Resonance elastic scattering ( ) A+1 X 0 Z Z * 1 n + A X 0 Z 1 A * A 1 A Fission Z 1 ( ) X 0 n + Z Inelastic scattering A+1 Z X + γ Radiative capture X Z X n 2 0

19 Nuclear Shell Model 1d 3/2 4 3s 3s 1/2 2 2d 1g 7/2 8 1d 5/2 6 1g 1g 9/2 10 2p 1/2 2 2p 1f 5/2 6 2p 3/2 4 1f 1f 7/ s 1d 1d 3/2 4 2s 1/2 2 1d 5/ p 1/2 2 1p 1p 3/ s 1s 1/2 2 2

20 Radiative capture The figure is for 238 U at E=6.67 ev. E 100 Incident neutron Neutron kinetic energy 6.67 ev 10 mc 2 + M 238 c U σ γ (E) (b) γ Cascade 239 U 92 Image by MIT OpenCourseWare. When the sum of the kinetic energy of the neutron in the CM and its binding energy correspond to an energy level of the compound nucleus, the neutron cross section exhibits a spike in its probability of interaction which are called resonances.

21 U C ross Sec tion (b) Energy (MeV) Image by MIT OpenCourseWare.

22 Cross Section Modeling Experimental data isn t available at every energy Quantum mechanical models are used to provide cross section values around data points Simplest version is Single Level Breit- Wigner Valid for widely spaced resonances

23 σ γ (E c ) = σ 0 Γ γ Γ ( E 0 Ec ( 1/ y 2, y = 2 Γ (E c _ E 0 ) σ max = σ 0 Γ γ Γ Γ = Total line width (FWHM) σ γ (E c ) 1 σmax 2 Γ Γ γ = Radiative line width σ 0 = Total cross section at E = E 0 E 0 E c Breit-Wigner Formula for Resonance Capture Cross Section Image by MIT OpenCourseWare.

24 Doppler Effect Cross sections are functions of relative speed between neutron and target nucleus Generally assumed that target is at rest Valid for smooth cross sections Not valid for resonances

25 Doppler Effect Resonances must be averaged over atom velocity Assume target nuclei have Maxwell - Boltzmann energy distribution As atom temperature increases Resonance becomes wider Resonance becomes shorter Area stays approximately the same

26 Maxwell-Boltzmann Distribution 100 o K 200 o K 400 o K Probability Velocity Image by MIT OpenCourseWare.

27 Doppler Effect σ ( E) σ (E, T) T 1 T 1 < T 2 <T 3 T 2 T 3 σ E 0 () σ( E v V ) E E Image by MIT OpenCourseWare.

28 Scattering - Inelastic Ec Neutron in A Z X * Inelastic scattering γ Emission mc 2 + M A c 2 A Z X σ in (E) M A+1 c 2 A+1 Z X Image by MIT OpenCourseWare. Inelastic scattering usually occurs for neutron energies above 10 kev. The excited state decays by gamma emission.

29 Resonance Scattering - Elastic E Neutron in Neutron out mc 2 + M A c 2 A Z X σ s (E) A+1 Z X Image by MIT OpenCourseWare. A compound nucleus is formed by the neutron and the nucleus. Peak and valley due to quantum mechanical interference term characterize the cross section. Kinetic energy is conserved.

30 Double-Differential Cross-Section v' v Image by MIT OpenCourseWare. The scattering cross-section will depend on both the energy and angle.

31 Scattering cross section - Double Differential E, Ω dω' E', Ω' ( ) ˆ ˆ ˆ σ E, Ω E', Ω' de'dω' [cm 2 s ] Image by MIT OpenCourseWare. σ s (E, Ωˆ E', Ωˆ ') [cm 2 / ev. sterradian] This characterizes neutron scattering from an incident energy E and direction Ω to a final energy E ' in the interval de ' and Ω ' in a solid angle dω '.

32 Lilley Neutron Scattering

33 Slowing Down Decrement Lilley 5.5.2

34 Lewis 3.3 Neutron Moderators

35 MIT OpenCourseWare Neutron Science and Reactor Physics Fall 2009 For information about citing these materials or our Terms of Use, visit: