Elements of NE midterm

rate of decay

dN/dt: rate of change of nuclei or rate of decay [nuclei/s]

λ: decay constant [s^-1]

N(t): number of nuclei present at time t [nuclei]

number of nuclei remaining

N(t): number of nuclei present at time t or number of nuclei remaining [nuclei]

N₀; initial number of nuclei

λ: decay constant [s^-1]

t: time [s]

number of atoms

N₀: initial number of atoms/nuclei [atoms] or [nuclei]

m: sample mass [g]

N_AV: avogadro’s number [6.022 × 10¹⁰ atoms/mol]

M; molar mass [g/mol]

activity

A=-dN/dt —> A=λN(t) —> A=A₀e^-λt

A: rate of decay [1bq=1decay/sec] or [1ci=3.7E10bq]

dN/dt: rate of change of nuclei [nuclei/s]

N(t): number of nuclei remaining

λ: decay constant [s^-1]

t: time [s]

specific activity

SA: specific activity [bq/g]

A: activity [bq]

m: sample mass [g]

half-life

t_1/2: half-life: time required for the radioactive nuclei to decay [s]

λ: decay constant [s^-1]

mean lifetime

mean lifetime: average lifetime of a nucleus [s]

production and decay

R: constant production rate [nuclei/s]

dN/dt: rate of change of nuclei or rate of decay [nuclei/s]

N(t): number of nuclei remaining

λ: decay constant [s^-1]

production and decay relationship

R > λN : production is greater than decay = N increases

R < λN : decay is greater than production= N decreases

R = λN : production is equal to decay = N stays constant

number of produced nuclei remaining

only valid when N(0)=0

R: constant production rate [nuclei/s]

N(t): number of nuclei remaining

λ: decay constant [s^-1]

t: time [s]

R/λ: max number of nuclei approached when production and decay balance

decay chain

a decay chain occurs when one isotope decays into another isotope, which can also be radioactive and decay again.

in the pic: the parent isotope ‘A’ decays and produces ‘B’ (daughter isotope) which also decays and produces D and E which can also be radioactive and can continue the chain

finding number of daughter nuclei remaining

we must first find N_A(t): since the parent nuclei A is only decaying, we can use the normal formula…

N_A(t)=N_A(0)e^-λ_At

then we can find N_B(t) under the conditions that N_B(0)=0 and λ_A ≠ λ_B

parent activity

daughter activity

secular equilibrium

decay constant of parent is much smaller than decay constant of daughter:

λ_A « λ_B

half-life of parent is much greater than half-life of B:

t_1/2,A»t_1/2,B

transient equilibrium

λ_A > λ_B

t_1/2,A < t_1/2,B

no equilibrium

λ_A ≅ λ_B

t_1/2,A ≅ t_1/2,B

in a question involving mass, number of atoms, activity, and specific activity

follow the sequence

if abundance (gamma symbol) is given

1] if given as percentage —> convert to decimal (divide by 100)

2] multiply it by the number of atoms formula (N₀)

neutron source

a material or nuclear reaction that produces free neutrons

when the reactor shuts down

neutrons are always present in a reactor core, even when the reactor is shut down.

why do we need neutron sources

neutron sources keep the neutron level high enough to be detected during reactor shutdown or start-up

which helps operators monitor the reactor safely

intrinsic sources

neutrons produced by materials already present in the reactor core for other purposes

example materials: fuel, moderator, burnable poision

installed sources

materials or components placed in the reactor specifically to produce neutrons

intrinsic source vs installed source

intrinsic sources already exist inside the reactor for another purpose while installed sources are deliberately added to produce neutrons

spontaneous fission

spontaneous fission occurs when a heavy nuclide splits without absorbing an incoming neutron

produces fission fragments & free neutrons

number of atoms per unit volume

“number of atoms” = N

“per unit volume“ = (per 1cm³)

the following equation represents atom density (number of atoms per unit volume = N)

N: atom density [ atoms / cm³ ]

ρ: density [ g / cm³ ]

N_AV: avogadro’s number [ 6.022 × 10¹⁰ ]

M: molar mass [g/mol]

density

nuclide density

a material can contain different isotopes of the same element

for example, uranium can contain both U-235 and U-238

each isotope has its own density

total number density

basically N = N₁ + N₂ + N₃ +…

N: total number density [ atoms / cm³ ]

N _i : number density of isotope i [ atoms / cm³ ]

average atomic mass

Ā: average atomic mass or average molar mass [ g/mol ]

A _i : molar mass of isotope i [ g/mol ]

N _i / N = enrichment percentage (convert to decimel)

the addition of the fraction (in decimal form) should equal 1 (100%)

microscopic cross section

microscopic cross section (σ): describes the likelihood of a neutron interacting with one nucleus [cm² or barns]

macroscopic cross section

macroscopic cross section (∑): describes the interaction probability per unit distance traveled through a material

∑ = Nσ

∑: macroscopic cross section [ cm^-1 ]

N: atom number density [ atoms / cm³ ]

σ: microscopic cross section [ cm² ] or [barns]

barns

1 barn = 10 ^-24 cm²

units of σ

macroscopic cross section of multiple different isotopes

(Total macroscopic cross section)

sum…

∑ = ∑_i N_i σ_i

f_i = N_i / N —> N_i = Nf_i

_{The parentheses is the average microscopic cross section $\overline{\sigma}=f_{i}\sigma$ (ex: $\overline{\sigma}$ $=$ $f_{235}\sigma_{235}$ $+$ $f_{238}\sigma_{238}$ )}

Average microscopic cross section

_{$\overline{\sigma}=f_{i}\sigma$}

_{(ex: $\overline{\sigma}$ $=$ $f_{235}\sigma_{235}$ $+$ $f_{238}\sigma_{238}$ )}

Average macroscopic cross section

$$\Sigma=N\overline{\sigma}$$

macroscopic cross section with volume fraction

macroscopic cross section with mass fraction

Steps to find the macroscopic cross section given enrichment

Step 1: calculate the average atomic mass

Step 2: calculate the total atomic number density

Step 3: calculate the average microscopic cross section

Step 4: calculate the macroscopic cross section