E64 · Structural photoemission

Photoelectron Structure: Direction and Depth

Acquire angular diffraction scans or standing-wave phase scans, then fit stored core-level counts and inspect the limits of structural inference.

Stored acquisition type $\text{—}$

Stored scan points $\text{—}$

Recorded events in core-level window $\text{—}$

Selected recorded coordinate $\text{—}$

Selected vacuum kinetic energy $\text{—}$

Photon energy from energy balance $\text{—}$

Fitted neighbour distance $\text{—}$

Fitted neighbour tilt $\text{—}$

Fitted coherent position, one period $\text{—}$

Equivalent depth within one period $\text{—}$

Fitted coherent fraction $\text{—}$

Fit relative RMS $\text{—}$

Experiment target $\text{—}$

Predict · acquire · fit

Ask what the scan constrains

Next acquisition: 0 diffraction, 1 standing wave

0

Assigned neighbour distance

r

2.8\,\mathring A

Assigned neighbour tilt from outward normal

\beta

45\,^\circ

Central vacuum kinetic energy

E_k

600\,eV

Calibrated angular acceptance RMS

\sigma_\theta

1\,^\circ

Signed angular half range

\theta_{\max}

70\,^\circ

Energy set: 1 central, 3 diverse

3

Assigned Gaussian layer centre depth

z_0

1.2\,nm

Assigned layer depth RMS width

\sigma_z

0.15\,nm

Calibrated fixed reflectivity

R

0.6

Ideal phase scan extent

\varphi_{\max}

360\,^\circ

Baseline expected counts per point

N_0

5000

Counts: 0 expectation, 1 seeded statistics

1

Selected recorded scan point

j

50\,\%

Display gain only

G

1

Diffraction target: three energies, broad angular coverage, narrow acceptance and sufficient counts. Standing-wave target: a broad phase scan with sufficient counts and substantial coherent modulation. A passed task does not certify a unique structure.

Controls plan the next acquisition; changing mode does not replace stored data. Acquire, then fit those counts. Presets load completed examples. Scan point and gain change presentation only.

Signed emission angles are measured from the outward surface normal. Layer depth is positive into the specimen. Standing-wave period: two nanometres; phase is an ideal calibrated input, not a simulated Bragg rocking curve.

Synthetic teaching models: one surface emitter and one scatterer with an assumed real scattering amplitude; or an ideal independently controlled standing-wave phase at fixed reflectivity. No element-specific phase shifts, multiple scattering, realistic Bragg rocking curve or nondipole correction. The layer signal is escape-weighted. Coherent position is periodic and is not a unique absolute depth. Apparatus, atomic spacing and layer thickness use different schematic magnifications.

Start with the essentials

Focus question: Which part of atomic structure can a count scan actually determine?
One-sentence intuition: A good fit answers the question posed by its measurement and assumptions.

Core mathematical model

Single-scattered amplitude

\begin{aligned}I(\theta)&=\left|1+\frac{f(\gamma)}{r}e^{ikr(1-\cos\gamma)}\right|^2\\\gamma&=\theta-\beta,\quad k=\sqrt{2m_eE_k}/\hbar\end{aligned}

The assumed scattering amplitude has a forward lobe; direct and scattered paths differ by the neighbour distance times the angular path factor.

Escape-weighted coherent moment

\begin{aligned}C&=\frac{\int_0^\infty\rho(z)e^{-z/\Lambda}\,e^{2\pi iz/d}\,dz}{\int_0^\infty\rho(z)e^{-z/\Lambda}\,dz}\\F&=|C|,\quad P=\frac{\arg C}{2\pi}\pmod 1\end{aligned}

The density is a surface-truncated Gaussian. The effective escape length includes the takeoff-angle cosine. Coherent position need not equal mean depth.

Ideal phase-dependent core-level yield

\begin{aligned}Y(\varphi)/A&=1+R\\&\quad+2\sqrt R\,F\cos(\varphi-2\pi P)\end{aligned}

Fit the constant, cosine and sine coefficients from stored counts with calibrated reflectivity. The amplitude is an unknown nuisance parameter.

Periodic inference

z_{\mathrm{equiv}}=dP,\qquad z_{\mathrm{equiv}}+nd\quad(n\in\mathbb Z)

Equivalent phase positions are separated by one period. Unknown layer abundance absorbs attenuation changes; one scan cannot select an absolute-depth branch.

Common difficulties

Scientific boundary

Typical misconceptionA low residual establishes a unique atomistic structure.

Better mental modelSynthetic teaching models: one surface emitter and one scatterer with an assumed real scattering amplitude; or an ideal independently controlled standing-wave phase at fixed reflectivity. No element-specific phase shifts, multiple scattering, realistic Bragg rocking curve or nondipole correction. The layer signal is escape-weighted. Coherent position is periodic and is not a unique absolute depth. Apparatus, atomic spacing and layer thickness use different schematic magnifications.

No synthetic confidence certificate

Typical misconceptionThe residual map is a confidence interval or experimental resolution.

Better mental modelThe unweighted least-squares map is descriptive. Angular acceptance is independently calibrated; energy-dependent scales are fitted. The standing-wave fit leaves noisy fractions above unity visible and rejects insufficient rank or grossly unphysical fits. All counts are synthetic; no uncertainty or external certification is claimed.

Run the experiment

01
Trace the two electron paths
Acquire an angular scan and move the selected point through the recorded angles.
What to observe: The collector follows the signed outward-normal angle. Direct and singly scattered waves interfere.
02
Fit the stored angular counts
Compare full coverage with the narrow and low-count examples.
What to observe: The search grid displays relative residual over distance and tilt. A sharp minimum is conditional on the assumed one-neighbour model.
03
Measure a layer with a standing wave
Acquire and fit a phase scan, then compare broad disorder and no reflection.
What to observe: Coherent modulation weakens when depths span multiple phases; zero reflection provides no phase information.
04
Challenge absolute-depth inference
Compare the completed phase scan with the one-period-deeper example.
What to observe: Phase repeats while absolute counts attenuate. Unknown abundance prevents a unique absolute depth; the two examples use separate synthetic specimens.