Slowing Down Theory

Abstract

In neutronic analysis for nuclear reactor systems, we look at three types of reactors depending upon the average energy of neutrons, which cause the bulk of the fission in the system:

Problems

Problem 6.1

An atom of uranium (U) with mass of 3.9529 × 10⁻²⁵ kg (i. e., m = 3.9529 × 10⁻²⁵ kg) at rest decays spontaneously into an atom of helium (He) mass of 3.8864 × 10⁻²⁷ kg (i.e., m = 3.8864 × 10⁻²⁷ kg). The helium atom is observed to move in the positive x-direction with velocity of 1.423 × 10⁷ m/sec Fig. 6.2. Do the following analyses as:

Fig. 6.2

(a) Before decay and (b) after decay

• (a) Find the velocity (i.e., magnitude and direction) of the thorium atom.

• (b) Find the total kinetic energy of the two atoms after the decay.

Problem 6.2

A helium atom (m = 6.6465 × 10⁻²⁷ kg) moving at a speed of V_He = 1.518 × 10⁶ m/s collides with an atom of nitrogen (m = 2.3253 × 10⁻²⁶ kg) at rest. After the collision, the helium atom is found to be moving with a velocity of ${\mathrm{V}}_{\mathrm{He}}^{\hbox{'}}$ = 1.199 × 10⁶ m/s at an angle of θ_He = 78.75^° relative to the direction of the original motion of the helium atom.

• (a) Find the velocity (magnitude and direction) of the nitrogen atom after the collision.

• (b) Compare the kinetic energy before the collision with the total kinetic energy of the atoms after the collision. Use Fig. 6.3 to solve this problem.

Fig. 6.3

(a) Before collision and (b) after collision

Problem 6.3

A particle of mass M is elastically scattered from a stationary proton of mass m. The proton is projected at an angle φ = 22.1^°, while the incident particle is scattered through an angle θ = 5.6^° with the incident direction. Calculate M in atomic mass units. (This event was recorded in photographic emulsions in the Wills Lab., Bristol.)

Problem 6.4

A particle of mass M is elastically scattered through an angle θ from a target particle of mass m initially at rest (M > m).

• (a) Show that the largest possible scattering angle θ_max in the lab. System is given by sinθ_max = m/M, the corresponding angle in the center of mass system (CMS) being cosθ^∗_max =  − m/M.

• (b) Further show that the maximum recoil angle for m is given by sinφ_max = [(M − m)/2M]^1/2.

• (c) Calculate the angle θ_max+φ_max for elastic collisions between the incident deuterons and target protons.

Problem 6.5

A deuteron of velocity u collides with another deuteron initially at rest. The collision results in the production of a proton and a triton (³H), the former moving at an angle 45^° with the direction of incidence. Assuming that this rearrangement collision may be approximated to an elastic collision (quasi-scattering), calculate the speed and direction of triton in the lab and center of mass (CM) system.

Problem 6.6

An α-particle from a radioactive source collides with a stationary proton and continues with a deflection of 10^°. Find the direction in which the proton moves (α-mass = 4.004 amu; proton mass = 1.008 amu).

Problem 6.7

An α-particle of kinetic energy of 20 MeV passes through a gas, and they are found to be elastically scattered at angles up to 30^° but not beyond. Explain this, and identify the gas. In what way, if any, does the limiting angle vary with energy?

Problem 6.8

A perfectly smooth sphere of mass m₁ moving with velocity ν collides elastically with a similar but initially stationary sphere of mass m₂ (m₁ > m₂) and is deflected through an angle θ_L. Describe how this collision would appear in the center of mass frame of reference and show that the relation between θ_L and the angle of deflection θ_M, in the center of mass frame is tanθ_L =  sin θ_M/[m₁/m₂+ cos θ_M]. Show also that θ_L cannot be greater than about 15^° if (m₁/m₂) = 4.

Problem 6.9

Show that the maximum velocity that can be imparted to a proton at rest by nonrelativistic alpha particle is 1.6 times the velocity of the incident alpha particle.

Problem 6.10

Show that the differential cross section σ(θ) for scattering of protons by protons in the lab system is related to σ(θ^∗) corresponding to the center of mass system (CMS) by the formula σ(θ) = 4 cos (θ^∗/2)σ(θ^∗).

Problem 6.11

If E₀ is the neutron energy and σ the total cross section for low energy n–p scattering assumed to be isotropic in the center of mass system (CMS), then show that in the laboratory system (LS), the proton energy distribution is given by (dσ_p/dE_p) = (σ/E₀) = constant.

Problem 6.12

Particles of mass m are elastically scattered off target nuclei of mass M initially at rest. Assuming that the scattering in the center of mass system (CMS) is isotropic shows that the angular distribution of M in the laboratory system (LS) has cosφ dependence.

Problem 6.13

A beam of particles of negligible size is elastically scattered from an infinitely heavy hard sphere of radius R. Assuming that the angle of reflection is equal to the angle of incidence in any encounter shows that σ(θ) is constant, that is, scattering is isotropic and the total cross section is equal to the geometric cross section, π/R².

Problem 6.14

Calculate the maximum wavelength of γ-rays which, in passing through matter, can lead to the creation of electrons.

Problem 6.15

A positron and an electron with negligible kinetic energy meet and annihilate one another, producing two γ -rays of equal energy. What is the wavelength of these γ-rays?

Problem 6.16

Show that electron-positron pair cannot be created by an isolated photon.

Problem 6.17

In dealing with diffusion of monoenergetic neutrons from a point source, the source diffusion equation for the flux distribution for a spherical symmetry is given in the following form:

$$ \frac{d^2\phi }{{d r}^2}+\frac{2}{r}\frac{d\phi}{d r}-{k}^2\phi =0 $$

(a)

Solve the differential equation (a) and show that the general solution is in form of the following type as:

$$ \phi (r)=A\left(\frac{e^{-\kappa r}}{r}\right) $$

(b)

where in Eq. (b), the parameter A can be evaluated based on what is so-called source condition. Hint: Assume ϕ(r) ≡ y/r.

Problem 6.18

Determine the value of parameter A, in Problem 6.17, if $\overline{\boldsymbol{J}}$ is the neutron current density at the surface of a sphere of radius r, with the source at the center. Utilize Fick’s law that you have learned in Chap. 3 in r-direction due to symmetrical condition. Assume the source strength is Q neutrons per second, i.e., to the number of neutrons emitted by the point source in all directions per second.

Problem 6.19

A hypothetical source of thermal neutrons emits 10⁶ neutrons per second into a surrounding “infinite” graphite block. Determine the neutron flux at distances of 27, 54, and 108 cm from this source. For graphite κ = 1/L is 0.0185 cm⁻¹ and the appropriate diffusion coefficient D is 0.94 cm. Hint: Use the result of the solution that you found in Problem 6.18.