E02 · Electron microscopy / foundations

Electron Gun Workbench

Which electron source gives the most coherent beam? A field emitter. Its tip emits from a tiny area, and the electrons leave with a narrow spread of energies. The price is a strong extraction field and ultrahigh vacuum. Choosing a source means trading coherence against beam current and how demanding the source is to run.

Compare four electron sources: tungsten, lanthanum hexaboride, Schottky and cold field emission. Choose an effective source diameter and the separation between two test points, and see how ideal fringe contrast trades against accepted source current. Then test whether you can get both while keeping a narrow energy width and adequate vacuum.

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

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

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

Ideal fringe visibility $0\,\mathrm{\%}$

More measurements and target

Emission source $\text{—}$

Relative emission against source preset $0$

First coherence zero at 0.1 m $0\,\mathrm{\mu m}$

Signed two-point coherence $0$

Emission model range $\text{—}$

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

Vacuum operating envelope $\text{—}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

Cathode temperature

T

2700\,\mathrm{K}

Extraction field

F

0.01\,\mathrm{GV/m}

Effective source diameter

d_s

20000\,\mathrm{nm}

Two-point baseline at 0.1 m

b

10\,\mathrm{\mu m}

Advanced settings

Work function

\phi

4.5\,\mathrm{eV}

Pressure exponent (base ten, Pa)

-4

Acceptance semi-angle

\alpha

10\,\mathrm{mrad}

Learning goal and model limits

Instrument settings

Electron Gun Workbench

At your chosen baseline, reach a positive coherence of at least 0.5 and a current of at least 10 nA. Keep the energy width at or below 0.8 eV, the vacuum adequate and the emission at least one tenth of the preset value, without leaving the range of the emission model. Then press Check target.

Model assumptions

Source parameters are representative values at 100 kV. The Richardson–Dushman law, Schottky barrier lowering and the elementary Fowler–Nordheim law set the relative emission. Brightness and energy widths are illustrative reference values, and the vacuum drift and Gaussian energy spectra are simplified teaching models. Coherence assumes a uniform disk of mutually incoherent emitters and 0.1 m of free propagation. The first zero is a defined baseline separation, not a measured property of the source. The fringes come from two ideal arms of equal intensity and equal path, with no time averaging. Partial coherence of real field-emission guns, space charge, detailed cathode chemistry and source lifetime are left out. Extreme extrapolations of emission are capped numerically and flagged. The accepted source current is a geometric emittance estimate, not the probe current at the specimen.

Physics tutorial

Which electron source gives the most coherent beam?

BackgroundCompare four electron sources: tungsten, lanthanum hexaboride, Schottky and cold field emission. Change the effective source size, the separation between two test points and the acceptance angle. Predict how ideal fringe contrast trades against accepted source current, then check the energy width and the vacuum.

Why it mattersBefore you trust an electron image, you need to know what the source, the lens column and the specimen itself allow.

Start with the essentials

Focus question: Which electron source gives the most coherent beam?
One-sentence intuition: A smaller source and a narrower energy width can improve coherence, but high extraction fields and vacuum requirements change what it takes to run the source.

Core mathematical model

Thermionic emission

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

The Lab uses this law 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 the barrier lowering in electronvolts. The thermionic exponent then uses the lowered work function.

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.

Ideal source current

I=\beta\,\pi(d_s/2)^2\,\pi\alpha^2

This is the current at the source plane, before any column aperture or demagnification. Brightness is taken as uniform over the source disk and the small acceptance cone.

Coherence from a uniform incoherent disk

\begin{aligned}\gamma(b)&=\frac{2J_1(u)}{u}\\u&=\frac{\pi d_s b}{\lambda L}\\b_0&=1.21967\frac{\lambda L}{d_s}\end{aligned}

The signed degree of coherence sets the contrast of ideal fringes from two equal arms, and its sign tells you when the fringes reverse phase. The first-zero baseline comes from the first positive zero of the Bessel function. This teaching assumption does not describe the intrinsic partial coherence of a real field-emission gun.

Common difficulties

Limits beyond brightness

Typical misconceptionThe brightest source always makes the smallest stable probe.

Better mental modelThe column optics, source emittance, energy width and vacuum stability also limit the probe.

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 energy width and a small virtual source.
02
Spoil the vacuum
Select cold FEG and raise the pressure exponent to minus four; watch the stability panel.
What to observe: A bright source can still be pushed outside its operating range.
03
Choose an operating point
Select Schottky FEG. Shrink the effective source and predict how much current you lose; increase the baseline and watch the fringe contrast fall. Aim for a positive coherence of at least 0.5, a current of at least 10 nA, an energy width of at most 0.8 eV, adequate vacuum and at least one tenth of the preset emission, without emission clipping. Then press Check target.
What to observe: Choosing a source means weighing energy width, coherence and vacuum together.