Formula sheet

Radiation, Plasma, and Charged-Particle Engineering Formula Sheet

Radiation and plasma formulas for dose, activity decay, shielding, detector dead time, photon flux, charged-particle motion, plasma frequency, Debye length, and validation.

Branch: Engineering Physics
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass equations used in radiation, plasma and charged-particle engineering systems. It covers dose, dose rate, activity decay, shielding attenuation, detector count-rate correction, photon flux, charged-particle motion, plasma frequency, Debye length, beam power, uncertainty and validation margins.

Use these equations for engineering screening, calibration review, detector-range checks, shielding estimates and physics interpretation. They are not a radiation-safety procedure, regulatory acceptance method, medical physics protocol, shielding certification or substitute for qualified site-specific review.

Constants and Notation

Symbol	Meaning	Typical unit
$D$	absorbed dose	Gy
$H$	equivalent or operational dose quantity	Sv
$\dot{H}$	dose-rate quantity	Sv/h or Sv/s
$E$	energy deposited or particle energy	J or eV
$m$	irradiated mass or observed count rate, by context	kg or s $^{-1}$
$n$	corrected event rate or number density, by context	s $^{-1}$ or m $^{-3}$
$b$	background count rate	s $^{-1}$
$\tau$	detector-chain dead time	s
$A$	activity or area, by context	Bq or m $^2$
$\lambda$	decay constant	s $^{-1}$
$t_{1/2}$	half-life	s
$\Phi$	fluence	particles/m $^2$
$\phi$	fluence rate or particle flux density	particles/(m $^2$ s)
$\mu$	linear attenuation coefficient	m $^{-1}$
$x$	shielding thickness	m
$B$	build-up factor or magnetic flux density, by context	dimensionless or T
$q$	particle charge	C
$v$	particle speed	m/s
$r_g$	gyroradius	m
$\omega_c$	gyro angular frequency	rad/s
$\omega_{pe}$	electron plasma angular frequency	rad/s
$\lambda_D$	Debye length	m
$n_e$	electron number density	m $^{-3}$
$T_e$	electron temperature	eV or K

Dose, Dose Rate and Exposure Time

Absorbed dose:

\displaystyle D=\frac{E_{dep}}{m}

Dose rate:

\displaystyle \dot{D}=\frac{D}{t}

Equivalent dose screen:

H=w_RD

where $w_R$ is a radiation weighting factor.

For a constant dose rate:

H=\dot{H}t

Use

Always state which dose quantity is being used. Absorbed dose, equivalent dose, ambient dose equivalent, personal dose equivalent and detector-indicated dose rate are not interchangeable without calibration, geometry and radiation-type assumptions.

Count Rate, Background and Dead Time

Background-subtracted observed rate:

m_{net}=m-b

First-pass nonparalyzable dead-time correction:

\displaystyle n=\frac{m}{1-m\tau}

Corrected net event rate:

n_{net}=n-b

Dead-time screen:

m\tau \ll 1

Dose-rate estimate from calibration coefficient $C_H$ :

\dot{H}=C_Hn_{net}

Use

This correction is a diagnostic screen. It does not validate pileup, baseline shift, discriminator walk, photodetector saturation, firmware filtering or pulse-height distortion. If $m\tau$ is not small, the detector chain needs range validation, not just arithmetic correction.

Activity Decay and Source Strength

Decay law:

A(t)=A_0e^{-\lambda t}

Half-life relation:

\displaystyle \lambda=\frac{\ln 2}{t_{1/2}}

Equivalent half-life form:

A(t)=A_0\,2^{-t/t_{1/2}}

Activity uncertainty from decay-time uncertainty:

\displaystyle \frac{u_A}{A}\approx \lambda u_t

when the initial activity uncertainty is treated separately.

Use

Source strength must be decay-corrected to the measurement time. Calibration records should state reference date, time standard, nuclide, emission spectrum, geometry and uncertainty.

Inverse-Square and Geometry Screens

For a point-like source in open geometry:

\displaystyle \dot{H}_2=\dot{H}_1\left(\frac{r_1}{r_2}\right)^2

For fluence from isotropic emission rate $S$ :

\displaystyle \phi(r)=\frac{S}{4\pi r^2}

Use

Inverse-square scaling is weak near extended sources, collimators, scattering surfaces, ducts, penetrations and shielding seams. Geometry should be measured, not guessed, before using distance scaling for release evidence.

Shielding Attenuation and Half-Value Layer

Exponential attenuation screen:

I=I_0e^{-\mu x}

With a build-up factor:

I=B_{build}I_0e^{-\mu x}

Half-value layer relation:

I=I_0\,2^{-x/HVL}

Transmission:

\displaystyle T=\frac{I}{I_0}

Use

These equations are first screens for narrow-beam or simplified shielding estimates. Real shielding review must account for spectrum, scatter, beam hardening, oblique paths, streaming gaps, penetrations, build-up, occupancy, controlled access and survey evidence.

Photon Energy, Flux and Beam Power

Photon energy:

\displaystyle E_{\gamma}=h\nu=\frac{hc}{\lambda}

For photon energy in electronvolts:

E_J=E_{eV}(1.602176634\times10^{-19})

Beam power from photon emission rate $\dot{N}$ :

P=\dot{N}E_{\gamma}

Flux density over area $A$ :

\displaystyle \phi=\frac{\dot{N}}{A}

Energy fluence rate:

\psi=\phi E_{\gamma}

Use

Photon flux does not equal dose without interaction coefficients, geometry, material, detector response and scattering assumptions. A high photon count can produce low absorbed dose in one material and a different dose in another.

Detector Statistics and Uncertainty

For counts $N$ with Poisson statistics:

u_N\approx \sqrt{N}

Count-rate standard uncertainty over counting time $T$ :

\displaystyle u_m=\frac{\sqrt{N}}{T}=\sqrt{\frac{m}{T}}

Relative counting uncertainty:

\displaystyle \frac{u_m}{m}\approx \frac{1}{\sqrt{N}}

Combined standard uncertainty for independent components:

u_c=\sqrt{\sum_i u_i^2}

Expanded uncertainty:

U=ku_c

Use

Counting uncertainty is often not the largest term. Calibration coefficient uncertainty, geometry, background subtraction, detector energy response, dead-time correction, survey positioning, environmental drift and instrument range can dominate.

Charged-Particle Motion in Fields

Lorentz force:

\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})

Kinetic energy from acceleration through potential:

E_k=qV

Nonrelativistic particle speed:

\displaystyle v=\sqrt{\frac{2E_k}{m}}

Gyro angular frequency:

\displaystyle \omega_c=\frac{|q|B}{m}

Gyrofrequency:

\displaystyle f_c=\frac{\omega_c}{2\pi}

Gyroradius:

\displaystyle r_g=\frac{v_\perp}{\omega_c}=\frac{mv_\perp}{|q|B}

Use

These formulas assume nonrelativistic speed, uniform magnetic field and simple perpendicular velocity. Electron beams, ion optics, plasma sheaths and accelerator components may require relativistic correction, space-charge modeling, collisions, fringe fields and electrode geometry.

Plasma Frequency and Debye Length

Electron plasma angular frequency:

\displaystyle \omega_{pe}=\sqrt{\frac{n_ee^2}{\epsilon_0m_e}}

Plasma frequency:

\displaystyle f_{pe}=\frac{\omega_{pe}}{2\pi}

Debye length with $T_e$ in joules:

\displaystyle \lambda_D=\sqrt{\frac{\epsilon_0 k_BT_e}{n_ee^2}}

Debye length with $T_e$ in electronvolts:

\displaystyle \lambda_D=\sqrt{\frac{\epsilon_0 T_{e,eV}}{n_ee}}

Approximate Debye-sphere population:

\displaystyle N_D=\frac{4}{3}\pi n_e\lambda_D^3

Use

Plasma formulas assume a plasma state that is meaningful: sufficient particles in a Debye sphere, quasi-neutrality away from sheaths, and compatible collision regime. Chamber pressure, gas species, power coupling, walls and magnetic field can change the plasma state strongly.

Beam Current, Particle Rate and Beam Power

Particle rate from current:

\displaystyle \dot{N}=\frac{I}{|q|}

Beam power for particles accelerated through voltage $V$ :

P=IV

Current density:

\displaystyle J=\frac{I}{A}

Heat flux if beam power is deposited over area $A$ :

\displaystyle q''=\frac{P}{A}

Use

Beam power can be small electrically and still create local thermal, charging, contamination or detector saturation problems if deposited on a small area. A beamline review should state where the energy is deposited and how the surface is cooled, grounded and monitored.

Worked Example 1: Detector Dead-Time and Dose Rate

A detector reports:

m=8000\ \text{s}^{-1}

Background is:

b=120\ \text{s}^{-1}

Dead time is:

\tau=12\ \mu\text{s}

Calibration coefficient is:

C_H=0.006\ \mu\text{Sv h}^{-1}/\text{s}^{-1}

Dead-time product:

m\tau=8000(12\times10^{-6})=0.096

Corrected event rate:

\displaystyle n=\frac{8000}{1-0.096}=8849.6\ \text{s}^{-1}

Corrected net rate:

n_{net}=8849.6-120=8729.6\ \text{s}^{-1}

Dose-rate estimate:

\dot{H}=0.006(8729.6)=52.4\ \mu\text{Sv/h}

Engineering comment: the dead-time product is below $0.1$ , so the correction is still a screening calculation rather than an obvious saturation failure. The result still needs calibration coefficient uncertainty, energy response, geometry and range validation before use in release evidence.

Worked Example 2: Shielding with Half-Value Layers

An unshielded survey point is estimated at:

\dot{H}_0=80\ \mu\text{Sv/h}

The selected shield has:

HVL=6\ \text{mm}

Thickness is:

x=24\ \text{mm}

Use build-up factor:

B_{build}=1.3

The number of half-value layers is:

\displaystyle \frac{x}{HVL}=\frac{24}{6}=4

Shielded dose-rate estimate:

\dot{H}=B_{build}\dot{H}_0 2^{-x/HVL}

\dot{H}=1.3(80)2^{-4}=6.5\ \mu\text{Sv/h}

Engineering comment: the simplified estimate is below a hypothetical $7.5\ \mu\text{Sv/h}$ project criterion, but the margin is narrow. A real release would require survey measurements at seams, penetrations, door overlap, beam-stop edges and worst-case operating settings.

Worked Example 3: Source Decay Correction

A source has activity:

A_0=120\ \text{MBq}

at the reference date. Half-life is:

t_{1/2}=73.8\ \text{days}

Estimate activity after:

t=180\ \text{days}

Use:

A(t)=A_0 2^{-t/t_{1/2}}

A(180)=120\,2^{-180/73.8}=22.1\ \text{MBq}

Engineering comment: the source is about $18\%$ of its reference activity after 180 days. Calibration or exposure planning based on the original activity would overstate output by more than a factor of five.

Worked Example 4: Photon Flux and Power

An x-ray source emits:

\dot{N}=2.0\times10^8\ \text{photons/s}

at photon energy:

E_{\gamma}=80\ \text{keV}

over area:

A=2.0\times10^{-4}\ \text{m}^2

Photon energy in joules:

E_J=80{,}000(1.602176634\times10^{-19})=1.28\times10^{-14}\ \text{J}

Beam power:

P=\dot{N}E_J=(2.0\times10^8)(1.28\times10^{-14})=2.56\times10^{-6}\ \text{W}

Flux density:

\displaystyle \phi=\frac{2.0\times10^8}{2.0\times10^{-4}}=1.0\times10^{12}\ \text{photons}/(\text{m}^2\text{s})

Engineering comment: photon power is very small in watts, but the radiation field can still be important because dose depends on energy deposition, material interaction and geometry. Electrical power and radiation hazard cannot be compared without an interaction model.

Worked Example 5: Plasma and Electron Gyromotion Screen

A low-pressure plasma has:

n_e=5.0\times10^{16}\ \text{m}^{-3}

T_e=3\ \text{eV}

Magnetic field:

B=0.020\ \text{T}

An electron has perpendicular kinetic energy:

E_k=100\ \text{eV}