Reactor Physics Flashcards
WhTs the best moderator to fuel ratio for water?
2:1
Why is water a good moderator?
Largest slowing down power due to small mass and large scattering xs, but it has a large thermal absorption xs
How are leakage and dimension related?
The larger the medium the less leakage. For an infinite medium you can assume no leakage.for a sphere leakage increases as r^2 and fission increases as r^3
What is the conversion or breeding ratio?
The average number of fissile atoms produced in a reactor per fuel atom consumed either by fission or absorption
What does it mean when a reactor breeds?
More fuel is produced than consumed, measured by breeding gain g
What values of eta allow a reactor to breed?
Much larger than 2. Note that eta increases with energy, so breeding reactors are often fast reactors
why do heavy nuclei not make good moderators?
take a large number of collisions to slow down since they have larger inelastic scattering cross sections
reaction rate
phi Sigma = nvN sigma
mean free path
average distance a neutron travels between collisions d = 1/Sigma_t
neutron lethargy
ln(E_0/E) where E_0 is 10 MeV
gain in lethargy
ln (E/E’) = 1 + alpha/(1-alpha)ln(alpha)
neutron generation lifetime
l = n(t)/L(t)
two factor formula
f eta = (nu Sigma_f^F)/(Sigma_a^F + Sigma_a^other)
six factor formula
fast fission factor - thermal + fast neutrons/thermal neutrons
resonance escape probability
thermal neutron utilization factor
thermal neutron reproduction factor
fast neutron non leakage probability
thermal neutron non leakage probability
what causes reactivity feedback caused by?
doppler broadening of resonances, changes in moderator density and temperature, changes in fission product inventory, changes due to core expansion
probability of decay between t and (t + dt)
p(t)dt = lambda e^{-lambda t} dt
binding energy
c^2[Nm_n + Zm_p - M_nucleus(A, Z)] in words strong nuclear force - surface tension binding + spin pairing + shell binding - coulomb repulsion
mass excess
[M_at(Z, N) - A]c^2
simple reactor power formula
P = epsilon_f R_f V = epsilon_f Sigma_f phi V
macroscopic cross section
Sigma = N sigma
Reaction Rate
R = phi Sigma = n v N sigma
intensity
I = I_0 e^{-Sigma_t x}
mean free path
lambda = 1 / Sigma_t
first collision probability
p(x)dx = Sigma_t e^{-Sigma_t x}
probability of uncollided flight
P = e^{- Sigma_t x}
thermal disadvantage factor
PhiMod_th/ PhiFuel_th
in hour equation
rho_0 = sl/(sl + 1) + 1/(sl +1) SUM((s beta_i)/(s + lambda_i))
general solution of in hour equation
phi(t) = A_1 e^{s_1t} + A_2 e^{s_2t} rho = 0, s1 = 0; rho goes to 1, s1 goes to infinity; rho goes to -infinity, s1 goes to -lambda
measure of reactivity in dollars
rho/beta
reactor period
T = 1/s1
prompt jump approximation
P1/P2 approx (beta - rho_2)/(beta - rho_1)
what will power look like after a long enough time?
P_2(t) = e^{-omega1 t}
Fick’s Law
J(r) = -D(r) del phi(r)
partial current J+
J+ = 1/4 phi(x) - 1/2D dphi/dx
when is the diffusion equation not valid?
- near boundaries where material properties change dramatically over the distance of several mean paths
- near localized sources
- in strongly absorbing media
fixed source diffusion equation
non multiplying medium with external source.
-D d^2phi(x)/dx^2 + Sigma_a phi(x) = S_ext(x)
eigenvalue problem diffusion equation
multiplying medium, no external source, homogeneous equation
-D d^2phi(x)/dx^2 + Sigma_a phi(x) = 1/k nu Sigma_f phi(x)
d^2phi(x)/dx^2 + B^2 phi(x) = 0 where B^2 = B^2_m/k
what is the finite flux boundary condition?
as x goes to infinity, flux stays finite.
solutions for 1d slab diffusion
- C1 e^{-kx} + C2e^{kx}
- C1 cosh (kx) + C2sinh(kx), k^2 <0
- C1 + C2x, k=0
- C1cos(kx) + C2sin(kx), k^2>0
Macroscopic cross section
probability per unit path length that a neutron will have a collision of a certain type. cm^-1
critical energy for fission reaction
minimum energy necessary to be supplied to a nucleus in order to deform the nucleus to a point where it can begin to split in two
products of a fission reaction that will be released up to a minute after
fission fragments, prompt neutrons, prompt gamma rays, delayed neutrons, delayed gamma rays, neutrinos
what’s the sequence of the life history of a neutron chart
P_FNL, p, P_TNL, P_AF, P_F
what are two assumptions in the six factor formula?
- two energy regions 2. neutrons emitted per fission is the same for both energy regions
heterogeneous thermal utilization factor
f = (Sigma_a V Phi)^F / [(Sigma_a V Phi)^F + (Sigma_a V Phi)^M]
thermal disadvantage factor
ratio of average thermal flux in moderator to average thermal neutron flux in fuel
thermal utilization factor, homogeneous
Sigma_a^F/(Sigma_a^F + Sigma_a^M)
which thermal utilization factor is larger, homogeneous or heterogeneous?
it depends on moderator to fuel ratio. if moderator volume is larger than fuel volume, the hetrogeneous factor will be smaller.
what is ell in the in hour equation?
prompt neutron lifetime,
what is beta in the in hour equation?
beta - delayed neutron fraction,
what is lambda in the in hour equation?
lambda - decay constant for delayed neutron precursors;
what is rho in the in hour equation?
rho - reactivity (k-1)/k,
what is s in the in hour equation?
s - reactor frequency and inverse reactor period.
what is a vacuum boundary condition mean?
incoming partial current is 0
what does a reflected boundary condition mean?
net current 0
what does an interface boundary condition mean?
current and flux are equal across boundary
what does a source condition mean?
lim as J goes to 0 = S/2
what is a symmetry boundary condition?
J(0) =0
what nuclide has highest binding energy per nucleon
Fe-56
what is the average binding energy per nucleon
8 MeV
Finding xs in 1/v region
sigma(E) = sigma(E’) E’/E
fast fission factor
epsilon
fast neutrons at all energies / fast neutrons at thermal energies
resonance escape probability
p,
number of neutrons that reach thermal energies / number of fast neutrons that start to slow down
thermal utilization factor
f
number of thermal neutrons absorbed in fuel / number of thermal neutrons absorbed in all material
thermal reproduction factor
eta
number of fast neutrons produced by thermal fission / number of thermal neutrons absorbed in nuclear fuel
Multigroup Diffusion Matrix form
M phi = 1/k F phi
removal cross section
Sigma_R,i = Sigma_t,i - Sigma, s, i to i
criticality condition
B_g^2 = (nu Sigma_f - Sigma_a)/D = B_m^2
important neutron poison
xe-135
why is xe-135 particularly terrible?
- its concentration can change dramatically over hours 2. the thermal absorption cross section is > 10^6
