E02 · Electron microscopy / foundations

Electron Gun Workbench

Compare tungsten, lanthanum hexaboride, Schottky and cold field emission. Tune temperature, work function, extraction field, virtual-source size and vacuum; watch energy width, accepted source current and stability change together. Choose a source with narrow energy spread and adequate vacuum.

Emission source $\text{—}$

Representative brightness $0\,\mathrm{A\,cm^{-2}\,sr^{-1}}$

Relative emission against source preset $0$

Energy width (FWHM) $0\,\mathrm{eV}$

Ideal accepted source current $0\,\mathrm{nA}$

Coherence scale at 0.1 m $0\,\mathrm{nm}$

Chamber pressure $0\,\mathrm{Pa}$

Vacuum operating envelope $\text{—}$

Experiment task $\text{—}$

Representative source parameters at 100 kV. Richardson–Dushman, Schottky barrier lowering and elementary Fowler–Nordheim laws scale relative emission. Brightness and energy widths use illustrative reference values; vacuum drift and Gaussian spectra are teaching surrogates. No space-charge, detailed cathode chemistry or lifetime prediction. Accepted source current is a geometrical emittance estimate, not specimen probe current.

Start with the essentials

Focus question: Which source can deliver a coherent, stable beam?
One-sentence intuition: A smaller source and a narrow energy width can improve coherence, while high extraction fields and vacuum requirements change the operating task.

Core mathematical model

Thermionic emission

J=A_RT^2\exp\!\left(-\frac{\phi}{k_BT}\right)

Used for relative emission, with a representative Richardson constant.

Schottky barrier lowering

\Delta\phi=\sqrt{\frac{eF}{4\pi\epsilon_0}}

The result in volts has the same numerical value as an electron energy in eV; use the lowered work function in the thermal exponent.

Elementary field emission

J\propto\frac{F^2}{\phi}\exp\!\left(-\frac{B_{\mathrm{FN}}\phi^{3/2}}F\right)

The elementary law omits image-barrier corrections and space charge.

Emittance and coherence estimates

I=\beta\,\pi(d_s/2)^2\,\pi\alpha^2,\qquad \ell_c\sim\frac{\lambda L}{\pi d_s}

The source-plane current estimate does not include column apertures or demagnification; the coherence scale uses a fixed propagation distance.

Common difficulties

Keep the model boundary explicit

Typical misconceptionThe brightest source always makes the smallest stable probe.

Better mental modelColumn optics, source emittance, energy width and vacuum stability are additional constraints.

Run the experiment

01
Compare sources
Predict which preset has the narrowest energy distribution, then compare all four.
What to observe: Cold field emission has a narrow representative width and small virtual source.
02
Lose the vacuum
Select cold FEG and raise the pressure exponent to minus four; watch the stability panel.
What to observe: A bright source can lie outside its operating envelope.
03
Choose an operating point
Reach energy width at most 0.8 eV, coherence scale at least 1000 nm, adequate vacuum and at least one tenth of the preset emission. Check target.
What to observe: Source selection is a joint energy, coherence and vacuum decision.