Exercise set

Plasma and Charged-Particle Beam Engineering Exercises

Solved plasma and charged-particle exercises for Debye length, plasma frequency, gyroradius, ion energy, beam power, heat flux, fluence and breakdown.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on plasma and charged-particle beam engineering: electric fields, ion energy, gyroradius, cyclotron frequency, Debye length, plasma frequency, beam current, beam power, heat flux, sheath exposure, fluence, ECR field and pressure-gap breakdown. Radiation dosimetry, shielding and detector counting are handled in a separate specialist exercise set.

Use the calculations as operating screens. Real plasma, accelerator, vacuum and beam systems require qualified equipment review, vacuum compatibility, high-voltage safety, interlocks, thermal limits, diagnostics, uncertainty analysis and approved operating procedures.

How to use these exercises

Treat each exercise as an operating envelope check. Start by identifying the system state: vacuum pumpdown, plasma ignition, steady recipe, pulsed beam operation, ECR tuning, sample exposure, thermal qualification, pressure-gap breakdown review or process release. The same current, voltage or fluence value can be acceptable in one mode and unsafe in another.

For each problem, separate:

plasma boundary: gas species, pressure, density, temperature, quasi-neutral region, sheath and diagnostic location;
beam boundary: current, energy, duty factor, pulse width, spot size, scan pattern, target material and cooling state;
high-voltage boundary: gap, edge fields, pressure drift, insulation, interlock state and stored energy;
release boundary: accept, derate, retune field, rescan beam, lower duty factor, improve cooling, repeat diagnostics or hold operation.

These exercises are intentionally first-pass. The useful habit is to connect a compact formula to the operating procedure, measurement boundary and failure mode that controls release.

Release Evidence Notes

Plasma and beam release evidence should state gas, pressure, gap, voltage, magnetic field, current, duty factor, beam area, pulse structure, substrate state, cooling limit, diagnostic calibration, interlock status and response if recipe or hardware conditions change. It should also state whether the evidence supports ignition, continuous operation, pulsed operation, recipe release, sample exposure, equipment qualification or safety hold.

For charged-particle processes, preserve both average and peak quantities. Average power can pass a cooling limit while pulse energy, peak current density or localized fluence damages a surface. Conversely, a safe peak value can still fail total dose, accumulated heat load or process-uniformity requirements.

Engineering Boundary Notes

The exercises use simplified single-particle and quasi-neutral plasma formulas. They do not replace plasma simulation, high-voltage qualification, vacuum design, accelerator safety analysis or process validation.

Single-particle motion formulas assume known fields and limited collision influence over the trajectory of interest. Plasma-scale formulas assume enough charged particles for collective behavior, a meaningful Debye length and diagnostic access to the relevant region. Breakdown screens are sensitive to gas, pressure, surface condition, electrode shape, contamination, edge field and transient pressure.

Beam fluence and heat-flux calculations assume that area, scan pattern and uniformity are known. A nominal beam diameter is not enough when edge under-dose, center hot spots, pulse overlap, raster gaps or target motion controls the release decision.

Common Release Mistakes

Common mistakes include treating average power as peak power, ignoring duty factor, applying plasma formulas when too few particles exist in a Debye sphere, using nominal gap voltage after pressure changes, and calculating fluence without beam uniformity evidence.

Other release mistakes include:

accepting ECR resonance from magnet current alone without a field map or plasma response evidence;
using chamber pressure as if it were local process pressure at the sheath or beam path;
comparing average heat flux with a target limit while pulse heating creates local thermal shock;
checking total charge while ignoring beam raster overlap, edge dose and target masking;
treating a guarded breakdown pass as stable when pressure drift, contamination or sharp electrodes can remove margin;
disabling interlocks for diagnostics without recording the temporary operating risk.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Particle motion and fields	1, 2, 3, 4, 14, 16	Energy, gyroradius, cyclotron and ECR field	Check trajectory and field coupling.
Plasma scale validity	5, 6, 7, 15	Debye length, plasma frequency, Debye population and mean free path	Decide whether the plasma model is credible.
Beam process release	8, 9, 10, 11, 12, 17, 18	Beam power, heat flux, sheath energy, fluence, duty and release gate	Accept, derate or hold the recipe.
Breakdown and interlocks	13, 18	Pressure-gap product and guarded voltage	Prevent ignition or flashover outside the approved state.

Exercise 1: Ion Energy from Bias Voltage

A singly charged ion falls through a 450 V sheath. Estimate ion energy in eV.

Solution

E=qV=450\ \text{eV}

Engineering Comment

Ion energy controls sputtering, implantation depth and substrate damage risk.

Plausibility Check

For a singly charged ion, volts and electron-volts have the same numerical value.

Exercise 2: Electric Field in a Gap

A 900 V potential is applied across a 12 mm gap. Find average electric field.

Solution

E=\dfrac{900}{0.012}=75000\ \text{V/m}

Engineering Comment

Local field enhancement at edges can exceed the average field.

Plausibility Check

About 1000 V over about 0.01 m gives order 100,000 V/m.

Exercise 3: Proton Gyroradius

A proton with perpendicular speed $2.0\times10^5$ m/s moves in a 0.08 T magnetic field. Find gyroradius.

Solution

r=\dfrac{mv}{qB}=\dfrac{1.67\times10^{-27}(2.0\times10^5)}{1.60\times10^{-19}(0.08)}=0.026\ \text{m}

Engineering Comment

Trajectory control fails when gyroradius is comparable to chamber or aperture dimensions.

Plausibility Check

Centimetre-scale radius is plausible for a proton in a modest magnetic field.

Exercise 4: Cyclotron Frequency

For the same proton and field, find cyclotron frequency.

Solution

f=\dfrac{qB}{2\pi m}=\dfrac{1.60\times10^{-19}(0.08)}{2\pi(1.67\times10^{-27})}=1.22\times10^6\ \text{Hz}

Engineering Comment

Cyclotron frequency sets resonance and timing scales for charged-particle motion.

Plausibility Check

The result is in the MHz range for protons at tenths of a tesla.

Exercise 5: Debye Length

Electron temperature is 3 eV and density is $1.0\times10^{16}$ m^-3. Use $\lambda_D=\sqrt{\epsilon_0 kT_e/(n_e e^2)}$ with $kT_e=3e$ joules.

Solution

\lambda_D=\sqrt{\dfrac{(8.85\times10^{-12})(3)(1.60\times10^{-19})}{(1.0\times10^{16})(1.60\times10^{-19})^2}}=1.29\times10^{-4}\ \text{m}

Engineering Comment

Diagnostic dimensions should be large compared with Debye length for bulk plasma assumptions.

Plausibility Check

Sub-millimetre Debye length is plausible for this density.

Exercise 6: Plasma Frequency

Find electron plasma frequency for $n_e=1.0\times10^{16}$ m^-3.

Solution

f_p=\dfrac{1}{2\pi}\sqrt{\dfrac{n_e e^2}{\epsilon_0 m_e}}=8.98\times10^8\ \text{Hz}

Engineering Comment

Plasma frequency helps decide whether fields penetrate or are screened.

Plausibility Check

The common estimate $f_p\approx 8.98\sqrt{n_e(\text{cm}^{-3})}$ kHz gives the same order.

Exercise 7: Debye Sphere Population

Using $n_e=1.0\times10^{16}$ m^-3 and $\lambda_D=1.29\times10^{-4}$ m, find particles in a Debye sphere.

Solution

N_D=\dfrac{4}{3}\pi n_e\lambda_D^3=90.0

Engineering Comment

Values much greater than 1 support a collective plasma interpretation.

Plausibility Check

The small Debye radius cubed is offset by high density, giving tens of particles.

Exercise 8: Ion Beam Power

An ion beam has current 12 mA and accelerating voltage 5 kV. Find beam power.

Solution

P=IV=0.012(5000)=60\ \text{W}

Engineering Comment

Beam power must be compared with target cooling and scan uniformity.

Plausibility Check

Milliamps times kilovolts gives watts to tens of watts.

Exercise 9: Beam Heat Flux

The 60 W beam strikes 20 cm2. Find average heat flux in W/cm2.

Solution

q''=\dfrac{60}{20}=3.0\ \text{W/cm}^2

Engineering Comment

Peak heat flux may be higher if the beam is nonuniform.

Plausibility Check

Sixty watts spread over twenty square centimetres gives three per square centimetre.

Exercise 10: Ion Rate

Beam current is 12 mA and ions are singly charged. Estimate ions per second.

Solution

\dot{N}=\dfrac{I}{e}=\dfrac{0.012}{1.60\times10^{-19}}=7.5\times10^{16}\ \text{ions/s}

Engineering Comment

Charge integration is often the most direct fluence basis.

Plausibility Check

One ampere is about $6.24\times10^{18}$ charges/s, so 0.012 A is about $7.5\times10^{16}$ .

Exercise 11: Fluence

The ion rate is $7.5\times10^{16}$ ions/s for 8 s over 20 cm2. Find fluence.

Solution

\Phi=\dfrac{7.5\times10^{16}(8)}{20}=3.0\times10^{16}\ \text{ions/cm}^2

Engineering Comment

Fluence release also needs beam-uniformity evidence.

Plausibility Check

Eight seconds gives $6.0\times10^{17}$ ions, divided by 20 cm2.

Exercise 12: Pulsed Average Power

A pulsed beam has peak power 2 kW and duty factor 0.04. Find average power.

Solution

P_{avg}=2000(0.04)=80\ \text{W}

Engineering Comment

Thermal design often depends on average power, but damage can depend on peak power.

Plausibility Check

Four percent of 2000 W is 80 W.

Exercise 13: Pressure-Gap Product

A gas gap is 4 mm at 80 Pa. Find pressure-gap product in Pa m.

Solution

pd=80(0.004)=0.32\ \text{Pa m}

Engineering Comment

Pressure-gap product is a first screen for breakdown behavior.

Plausibility Check

Small gap and low pressure give a value below 1 Pa m.

Exercise 14: Guarded Breakdown Voltage

A breakdown screen gives 950 V at the current pressure-gap product. Applied voltage is 820 V and uncertainty allowance is 80 V. Check guarded margin.

Solution

V_g=820+80=900\ \text{V},\qquad M=950-900=50\ \text{V}

Engineering Comment

The pass is narrow; pressure drift or edge fields could remove the margin.

Plausibility Check

Guarded applied voltage remains below the screen by 50 V.

Exercise 15: Electron Cyclotron Resonance Field

Find magnetic field for 2.45 GHz electron cyclotron resonance using $f=eB/(2\pi m_e)$ .

Solution

B=\dfrac{2\pi m_e f}{e}=\dfrac{2\pi(9.11\times10^{-31})(2.45\times10^9)}{1.60\times10^{-19}}=0.0876\ \text{T}

Engineering Comment

ECR matching needs field map evidence, not only nominal magnet current.

Plausibility Check

2.45 GHz ECR is known to be near 0.0875 T.

Exercise 16: Mean Free Path Screen

Neutral density is $2.0\times10^{20}$ m^-3 and collision cross-section is $3.0\times10^{-19}$ m2. Estimate mean free path.

Solution

\lambda=\dfrac{1}{n\sigma}=\dfrac{1}{(2.0\times10^{20})(3.0\times10^{-19})}=0.0167\ \text{m}

Engineering Comment

Mean free path should be compared with sheath, gap and chamber dimensions.

Plausibility Check

The product $n\sigma$ is 60 m^-1, so the path is about 1/60 m.

Exercise 17: Fluence Uniformity Gate

Required fluence window is $(3.0\pm0.3)\times10^{16}$ ions/cm2. Measured center is $3.1\times10^{16}$ and edge is $2.6\times10^{16}$ . Decide.

Solution

Allowed range:

2.7\times10^{16}\le\Phi\le3.3\times10^{16}

The edge fails because $2.6\times10^{16}$ is below the lower limit.

Engineering Comment

Average fluence is not enough when edge under-dose controls process release.

Plausibility Check

The center passes and the edge is visibly below the allowed lower bound.

Exercise 18: Plasma Process Release Gate

Release requires guarded breakdown pass, heat flux below 4 W/cm2 and fluence uniformity pass. Results are pass, 3.0 W/cm2 and fail. Decide.

Solution

\text{breakdown}=\text{pass},\qquad 3.0<4.0,\qquad \text{fluence}=\text{fail}

The process release fails because fluence uniformity fails.

Engineering Comment

Electrical and thermal checks cannot compensate for a nonuniform beam process.

Plausibility Check

One required release criterion fails, so the process must be held.

Validation Package Checklist

gas, pressure, gap, voltage and field basis are stated;
beam current, pulse timing, area and duty factor are measured or controlled;
plasma scale checks support the chosen model;
thermal limits use both average and peak exposure where relevant;
fluence includes uniformity, not only total charge;
magnetic-field map, ECR tuning evidence or trajectory evidence is included when field coupling controls the result;
pressure measurement location, gas composition, pump state and leak/outgassing condition are documented;
interlock, enclosure, grounding, stored-energy and high-voltage discharge controls are verified before operation;
diagnostics have calibration or cross-check evidence at the relevant bandwidth, pulse timing and spatial location;
release decision states accept, derate, rescan, adjust recipe, retune field, repeat diagnostics, restore interlock or hold.

The final release statement should identify the controlling gate. A process can pass breakdown and heat-flux checks while failing fluence uniformity, or pass plasma-scale checks while high-voltage safety evidence still blocks operation.

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