1 NUEN 301 EXAM 1, September 23, 2008 No Calculators!!! [points] 1. A mono-energetic, mono-directional neutron beam of intensity I0 = 3×106 n/cm2-s is incident upon a homogeneous cubical target. The target, which has width w = 5 cm, is NOT “thin.” [This means you cannot ignore attenuation of the beam.] The incident beam is perpendicular to one face of the cube and it completely covers this face. The neutrons in the beam are traveling in the +ex direction, which is toward the right in the figure below. I0ncm2− s⎡⎣⎢⎤⎦⎥ The material in the cube does not scatter neutrons and does not fission – its only interaction with neutrons is radiative capture. Σt = Σcapture = 0.6 cm−1. In the following, be careful to state what is a vector and what is a scalar, and make sure that your notation clearly indicates which is which! [5] a) What is the intensity of the beam after it passes through the cube? Give an expression that could be punched into a calculator. [5] b) How many neutrons per second enter the cube, and how many exit? Give an expression that could be punched into a calculator. [5] c) What is the neutron net current density at x=w/2 (in the center of the cube)? Be careful with notation and details! 2. What is the physical meaning of: [10] a) the total (not energy-dependent) scalar flux integrated over a volume? [“the rate at which ...] [5] b) the total scalar flux integrated over a surface? [5] c) the quantity en•J, where J is total net current density, integrated over a surface that is perpendicular to the unit vector en? 3. In a certain thermal reactor, for every 100 neutrons emitted from fission, 10 cause fission while still “fast” and 30 cause fission after becoming “thermal.” The other 60 either leak or get absorbed without causing fission. νT is 2.49 and νF is 2.53. [10] a) What is the multiplication factor, or is there insufficient information to calculate it? [5] b) What is the fast-fission factor, or is there insufficient information to calculate it?2 4. Consider a homogeneous cubical reactor: –w/2 = x = w/2 , – w/2 = y = w/2 , –w/2 = z = w/2 . You are given φx, y, z( )= Acosπxw⎛⎝⎜⎞⎠⎟, A>0, , Jyx, y, z( )= Jzx, y, z( )= 0, Σt = 2 cm–1, Σa = 0.9 cm–1, Σf = 0.4 cm–1, ν = 2.5 n/fission. [20] a) Calculate the production rate, absorption rate, and net outleakage rate (n/s) for this reactor. Your expression for each rate should be in terms of the constant A and the cube width w. [20] b) Find the width, w, that would make this reactor critical, or convince me that no such width exists. If you have an answer for w, it should be in a form that I can punch into a calculator. [10] 5. In the middle of a large homogeneous reactor, the energy-dependent scalar flux is roughly independent of position. Suppose that in such a reactor the energy-dependent scalar flux is approximately: φ(E) = [4×109 n/cm2-s] / E , for 1 eV < E < 1 MeV, and the differential scattering cross section is ΣsEi→ Ef( )=0, Ef> Ei,0.5 cm−1⎡⎣⎤⎦Ei, Ef∈ 0, Ei( ),⎧⎨⎪⎪⎩⎪⎪⎫⎬⎪⎪⎭⎪⎪ . What is the rate density (scatters per cm3 per s) at which neutrons scatter directly from energies above 100 keV to energies below 5 keV? Your answer should be a number or an arrangement of numbers that I can punch into a calculator. Extra-credit problem on next page ...3 Extra Credit Two mono-energetic, mono-directional neutron beams are incident upon a homogeneous slab that is infinite in the y and z directions. The beams completely cover the portion of the target that we are interested in. See Figure. Given the energy of the incident neutrons, the material cross sections are: [5] What is the scalar flux as a function of x in this