Reactor Physics Test 1

Neutrons are distributed in the Maxwell-Boltzmann distribution given by M(E)=2π/(πkT)^3/2 E^1/2 exp (-E/kT) show 0∫∞ M(E)dE = 1 from the formula 0∫∞ x^2 exp(-x^2)dx = π^1/2/4

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N

atom density, number of molecules or atoms

g

constant known as the statistical factor

E

energy

Why neutron is so important in a nuclear fission reactor?

it makes fission easier

λ

mean free path

σ

microscopic cross-section

M

molecular mass

φ

neutron flux

Γn

neutron line width = √E

v

number of neutrons emitted per fusion

η

number of neutrons emitted per neutron absorbed, reproduction factor

A small indium foil is placed at a point in a reactor where the 2,200 meters-per second flux is 5 X 1012 neutrons/cm2-sec. The neutron density can be represented by a Maxwellian function with a temperature of 600°C. At what rate are the neutrons absorbed per cm3 in the foil?

picture #2

Γγ

radiation width

σy

radioactive capture

R

reaction rate

Er

the resonance energy= energy of the lowest energy resonance

Σ

total macroscopic cross section

A boiling water reactor operates at 1000 psi. At that pressure the density of water and of steam are .74 gm/cm^3 and .036 gm/cm^3. The microscopic cross section of H and O are 38b and 4.2×10^-5b, What is the macroscopic total cross section of the water and steam?

w=1.88cm-1 s=.0914cm-1 (σ=2×38×10^-24+4.2×10^-29=76×10^-24cm2 Σ=pN(2σH + σO)/18 Σw=.74×6.023×10^24×76×10^-24/18 Σs=.036×6.023×10^24×76×10^-24/18)

Yr

wavelength of neutrons with energy Er

Which fuel is best for thermal reactors?

Lower enriched fuel

An isotope of plutonium has a thermal fission cross section of 747.40 barns, a thermal capture cross section of 270.33 barns, and a thermal scattering cross section of 7.7 barns. What is the thermal absorption cross section of this isotope?

1017.73 barns (σa = σf + σc = 747.40 barns + 270.33 barns)

Determine the fraction of fission neutrons born with energy of less than .1 Mev. X(E)=.453exp(-1.036E)sinh(E2.29^1/2)

.014 (0∫.1 equation)

The value of σy for 1H at .0253 eV is .332 b. What is σy at 1 eV?

.0528b (σy sqrt(E0/E) = .332sqrt(.0253/1))

Suppose that a fuel rod in a reactor consists of 20% enriched UO2 (U-235+U-238). If this reactor runs at 0.1 eV or 1 Mev, what is the value of η?

.1=2.035 1M=1.385 (η=ev(ε)σᶠ(ε)+(1-e)v(E)σᶠ(E)/eσa(ε)+(1-e)σa(E) .2×2.43×255+0/.2(255+44)+.8(1.36) .2×1.904×2.65+.8×2.55×.0428/.2(1.904+.5555)+.8(.0428+.3319))

Find the macroscopic thermal neutron absorption cross section (cm-1) for iron, which has a density of 7.86 g/cm3. The microscopic cross section for absorption of iron is 2.56 barns and the gram atomic weight is 55.847 g.

.217 cm-1 (N=PNa/M = 7.86×6.022×10^23/55.847 = 8.48×10^22 | Σ=Nσa = 8.48×10^22×2.56)

Equal volumes of graphite and iron are mixed together .15% of the volume of the resulting mixture is occupied by air pockets. Find the total macroscopic cross section given the following data: σC=4.75lb, σFe=10.9lb, ρC=1.6gm/cm3, ρFe=7.7gm/cm3.

.555cm-1 (VFe+Vc/V=.85=VFe/V Vc/V=.425 Σ=VepNσ/VA (.425×7.87×6.023×10^23×10.9×10^-24/55.85 + .425×1.6×6.023×10^23×4.75×10^-24/12.01))

A 1 -MeV neutron is scattered through an angle of 45° in a collision with a 2H nucleus. (a) What is the energy of the scattered neutron? (b) What is the energy of the recoiling nucleus?

.738 MeV and .262 MeV (E'=1/(2+1)^2 (cos45+sqrt(4-sin^245))^2 Ea=1-.723)

We want to evaluate ξH2O when σH=20b and σ0=3.8b

.924 (ξO=2/(16+.66)=.12 ξH=1 (2×1×20+.12×3.8/2×20+3.8))

Calculate the reproduction factor for a reactor that uses 10% enriched uranium fuel. The microscopic absorption cross section for uranium-235 is 694 barns. The microscopic absorption cross section for uranium-238 is 2.71 barns. The microscopic fission cross section for uranium-235 is 582 barns. The atom density of uranium-238 is 4.35 x 10^22 atoms/cm3. The atom density of uranium-235 is 4.83 x 10^21 atoms/cm3. v is 2.42.

1.96 (η=Nσv/Nσa+Nσa (4.83×10^21×582×10^-24×2.42/4.83×10^21×694×10^-24 + 4.35×10^22×2.71×10^-24))

What is the total macroscopic thermal cross section (m-1) of uranium dioxide (UO2) that has been enriched to 4%?Assume, σ235=607.5lb, σ238=11.8lb, σO=3.8lb, and that UO2 has a density of 10.5gm/cm3?

101.3 m-1 (σU=eσU235+(1-e)σU238 (.04×607.5 + .96×11.8=35.63b) σUO2=σU + 2σO (35.63 + 2×3.8=43.23b) N=PL/M (10500×6.023×10^23/(238 + 2×16)10^-3 = 2.343×10^28 m-3) Σ=Nσ (2.343×10^28×43.23×10^-28))

Suppose that a fuel rod in a thermal water reactor consists of 10% enriched UO2. What is the value of η? (U235 v=2.43 σf=580 σc=107) (U238 v=dne σf=0 σc=2.75)

2.0567 (η=evᶠⁱσᶠⁱ+(1-e)vᶠᵉσᶠᵉ/eσaᶠⁱ+(1-e)σaᶠᵉ (e×2.43×580 + (1-e)×0/e(580+107)+(1-e)(2.75+0))

What is the average number of elastic scattering collisions needed to slow a neutron down from 2 MeV to 0.025 eV in the Deuterium (A=2)?

24.3 (ξ=2/2+.66=.75 n=(1/.75 ln(2×10^6/.025)))

An isotope of plutonium has a thermal fission cross section of 747.40 barns, a thermal capture cross section of 270.33 barns, and a thermal scattering cross section of 7.7 barns. What is the probability that when the isotope absorbs a thermal neutron the nucleus will not split apart into two pieces,?

26.6% (σc/σa = 270.33/1017.73 = 0.266)

Using the Breit-Wigner formula. Derive the equation for σY as a function of E in the vicinity of the first resonance in U-238, which occurs at an energy of 6.67 eV (Er) . The parameters of this resonance are: γr = 1.107 X 10-9 cm, Γn =1.52 meV (meV = millielectron volts), Γg = 26 meV and g = 1.

3.85×10^-24/(E-6.67)^2+1.89×10^-4 ((1.107×10^-9)^2/4π 1.52×10^-3×26×10^-3/(E-6.67)^2+(27.52×10^-3)^2/4)

Calculate the microscopic absorbtion cross section for .025 eV neutrons of natural uranium, which consiosts of 99.285% U-238 and .715% U-235. The microscopic cross-sections for .025 eV neutrons are (U-238, captur\e cross section)=> 2.27 barns. (U-238, fission cross section)=> 0 barns. (U-235, capture cross section)=> 101 barns. (U-235, fission cross section)=> 579 barns.

7.56 barns (σa = 0.99285× (2.72 + 0) + 0.00715× (101+ 579)

An isotope of plutonium has a thermal fission cross section of 747.40 barns, a thermal capture cross section of 270.33 barns, and a thermal scattering cross section of 7.7 barns. What is the probability that when the isotope absorbs a thermal neutron the nucleus will split apart into two pieces?

73.4% (σf/σa = 747.4/1017.73 = 0.734)

Calculate the mean free paths (in cm) for absorption for the following scenario:An alloy is composed of 95% aluminum and 5% silicon (by weight). The density of the alloy is 2.66 g/cm3. (aluminum weight= 26.9815) (silicon weight= 28.0855) (aluminum σa=.23) (silicon σa=.16)

74.3 cm (N=PNa/M (.95×2.66×6.022×10^23/26.9815= 5.64×10^22) (.05×2.66×6.022×10^23/28.0855= 2.85×10^21) Σ=Nσa (5.64×10^22×.23×10^-24 + 2.85×10^21×.16×10^-24= .0134) λ=1/Σ (1/.01345))

What is the minimum number of elastic scattering collisions for a neutron from 2 Mev to 0.025 eV in Deuterium (A=2)?

8.3 (2ln(3)=2.197 n=1/2.197 ln(2×10^6/.025))

Which fuel is best for fast reactors?

High enriched fuel

How does enrichment impact η?

The number of higher enriched neutrons admitted have a higher chance of being absorbed in the absorbtion cross section

Why reactors are designed either to operate with a fast neutron spectrum (E > 0.1 MeV) or a thermal spectrum (E < 0.1 eV)?

The renassance region is uncertain

Wwhy does fast reactors need fuels enriched to more than 10%, although η increases significantly after 1 MeV?

Too many neutrons are scattered to a low energy range, higher the enrichment to reduce loss