Nuclear Physics Formulas

Complete collection of nuclear physics formulas with detailed explanations.

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Nuclear Decay

Radioactive Decay

N(t) = N_0 e^{-\lambda t}

Number of nuclei at time t equals initial number times exponential of negative decay constant times time.

Notation:

N(t):Number of nuclei at time t

N₀:Initial number of nuclei

λ:Decay constant

t:Time

Units:

N(t), N₀:dimensionless

λ:s⁻¹

t:s

Applications:

•Carbon dating
•Nuclear medicine
•Radiation safety

Limitations:

Exponential decay

Half-Life

T_{1/2} = \frac{\ln(2)}{\lambda}

Half-life equals natural logarithm of 2 divided by decay constant.

Notation:

T₁/₂:Half-life

λ:Decay constant

Units:

T₁/₂:s

λ:s⁻¹

Applications:

•Nuclear dating
•Medical isotopes
•Nuclear waste

Limitations:

First-order decay

Decay Rate

A = \lambda N

Activity equals decay constant times number of nuclei.

Notation:

A:Activity

λ:Decay constant

N:Number of nuclei

Units:

A:Bq (becquerels)

λ:s⁻¹

N:dimensionless

Applications:

•Radiation measurement
•Nuclear power
•Medical imaging

Limitations:

First-order decay

Mean Lifetime

\tau = \frac{1}{\lambda}

Mean lifetime equals reciprocal of decay constant.

Notation:

τ:Mean lifetime

λ:Decay constant

Units:

τ:s

λ:s⁻¹

Applications:

•Nuclear stability
•Decay calculations
•Particle physics

Limitations:

Exponential decay

⚛️

Nuclear Reactions

Mass-Energy Equivalence

E = mc^2

Energy equals mass times speed of light squared.

Notation:

E:Energy

m:Mass

c:Speed of light

Units:

E:J

m:kg

c:3 × 10⁸ m/s

Applications:

•Nuclear fusion
•Nuclear fission
•Particle physics

Limitations:

Rest mass energy

Binding Energy

E_b = \Delta mc^2

Binding energy equals mass defect times speed of light squared.

Notation:

E_b:Binding energy

Δm:Mass defect

c:Speed of light

Units:

E_b:J

Δm:kg

c:3 × 10⁸ m/s

Applications:

•Nuclear stability
•Fusion reactions
•Nuclear power

Limitations:

Rest mass approximation

Q-Value

Q = (m_{initial} - m_{final})c^2

Q-value equals initial mass minus final mass times speed of light squared.

Notation:

Q:Q-value

m_initial:Initial mass

m_final:Final mass

c:Speed of light

Units:

Q:J

m_initial, m_final:kg

c:3 × 10⁸ m/s

Applications:

•Nuclear reactions
•Energy release
•Reaction feasibility

Limitations:

Rest mass approximation

Nuclear Cross Section

\sigma = \frac{R}{\Phi n}

Cross section equals reaction rate divided by flux times target density.

Notation:

σ:Cross section

R:Reaction rate

Φ:Particle flux

n:Target density

Units:

σ:barn (10⁻²⁸ m²)

R:s⁻¹

Φ:m⁻²s⁻¹

n:m⁻³

Applications:

•Nuclear scattering
•Reaction rates
•Neutron physics

Limitations:

Point-like particles

💥

Fission and Fusion

Fission Energy

E_{fission} = (m_U - m_{fragments})c^2

Fission energy equals uranium mass minus fragment masses times speed of light squared.

Notation:

E_fission:Fission energy

m_U:Uranium mass

m_fragments:Fragment masses

c:Speed of light

Units:

E_fission:J

m_U, m_fragments:kg

c:3 × 10⁸ m/s

Applications:

•Nuclear power plants
•Atomic bombs
•Energy production

Limitations:

Rest mass approximation

Fusion Energy

E_{fusion} = (m_{reactants} - m_{products})c^2

Fusion energy equals reactant masses minus product masses times speed of light squared.

Notation:

E_fusion:Fusion energy

m_reactants:Reactant masses

m_products:Product masses

c:Speed of light

Units:

E_fusion:J

m_reactants, m_products:kg

c:3 × 10⁸ m/s

Applications:

•Hydrogen bombs
•Fusion reactors
•Stellar energy

Limitations:

Rest mass approximation

Critical Mass

M_{critical} = \frac{\pi^2 D^2}{\nu\Sigma_f}

Critical mass equals π squared times diffusion coefficient squared divided by neutron velocity times fission cross section.

Notation:

M_critical:Critical mass

D:Diffusion coefficient

ν:Neutron velocity

Σ_f:Fission cross section

Units:

M_critical:kg

D:m

ν:m/s

Σ_f:m⁻¹

Applications:

•Nuclear weapons
•Reactor design
•Criticality safety

Limitations:

Spherical geometry

🛡️

Radiation Protection

Dose Rate

\dot{D} = \frac{A\Gamma}{d^2}

Dose rate equals activity times gamma constant divided by distance squared.

Notation:

Ḋ:Dose rate

A:Activity

Γ:Gamma constant

d:Distance

Units:

Ḋ:Gy/h

A:Bq

Γ:Gy·m²/(h·Bq)

d:m

Applications:

•Radiation safety
•Shielding design
•Dose calculations

Limitations:

Point source

Half-Value Layer

HVL = \frac{\ln(2)}{\mu}

Half-value layer equals natural logarithm of 2 divided by linear attenuation coefficient.

Notation:

HVL:Half-value layer

μ:Linear attenuation coefficient

Units:

HVL:m

μ:m⁻¹

Applications:

•Shielding design
•Radiation protection
•Material selection

Limitations:

Narrow beam geometry

Exposure Rate

\dot{X} = \frac{A\Gamma}{d^2}

Exposure rate equals activity times gamma constant divided by distance squared.

Notation:

Ẋ:Exposure rate

A:Activity

Γ:Gamma constant

d:Distance

Units:

Ẋ:R/h

A:Ci

Γ:R·m²/(h·Ci)

d:m

Applications:

•Radiation monitoring
•Safety assessments
•Regulatory compliance

Limitations:

Point source