= Solution
<Atomic force microscopy> senses tip–sample interactions through a compliant cantilever. Contact deflection, oscillation amplitude, phase or resonance <frequency> can be converted into a feedback signal. Chemical bonding and short-range repulsion, van der Waals <forces>, electrostatic <forces> and, where relevant, magnetic or capillary interactions produce contrast. Apparent height at a fixed <force> or <frequency> setpoint can therefore contain interaction contrast as well as geometry. Lateral resolution depends on the effective tip apex and interaction range; finite tip radius broadens features. Vertical sensitivity also depends on thermal and detector noise, cantilever calibration, vibration, drift and feedback bandwidth. Strongly localized short-range <forces> favor atomic resolution, whereas extended interactions average over more of the tip and surface.
For small oscillations about equilibrium separation $z_0$, write $z=z_0+x$ and linearize $F(z)=F(z_0)+F'(z_0)x$. The constant <force> is absorbed into the equilibrium displacement. The <cantilever> equation becomes
$$
m_{\rm eff}\ddot x=-\left[c-F'(z_0)\right]x,\qquad
\omega^2=\omega_0^2\left[1-\frac{F'(z_0)}c\right],\quad\omega_0^2=c/m_{\rm eff}.
$$
Consequently the <small-amplitude AFM frequency shift> is
$$
\boxed{\frac{\Delta\omega}{\omega_0}=\sqrt{1-\frac{F'(z_0)}c}-1
\simeq-\frac1{2c}\frac{dF}{dz}(z_0).}
$$
The linear expression is the small-force-gradient approximation, $|F'|\ll c$, in addition to the small-oscillation assumption. For an attractive <force> $F=-C/z^2$, $F'>0$ and the resonance softens. A finite oscillation amplitude samples a weighted range of <forces> rather than only the gradient at one separation, and $F'\ge c$ destroys local stability in this simple static model.
\Image[/past-exam-of-the-frontiers-of-experimental-condensed-matter-physics-course-of-the-university-of-cambridge/2013/paper-1-afm-frequency-gradient.png]
{title=AFM frequency shift: exact local linearized oscillator and the weak-gradient approximation}
In a <scanning electron microscope>, a focused <electron> beam scans the surface and a detector records an emitted signal at each beam position. Low-energy <secondary electrons> escape from near the surface; changes in local slope, edges and detector-facing geometry give strong <secondary-electron topographic contrast>. <Backscattered electrons> result from elastic deflection of primary <electrons> and generally have a larger yield for higher atomic number. Their <backscattered-electron atomic-number contrast> therefore helps separate compositional regions. Detector geometry, crystal orientation and topography can also affect this signal, so brightness is not a unique elemental identification.
An <environmental scanning electron microscope> admits gas around the specimen while pressure-limiting apertures and differential pumping preserve the electron-optical column's vacuum. <Electron> collisions ionize the gas. Positive ions can neutralize negative charging of a poorly conducting specimen, enabling observation without a conductive coating. A <gaseous secondary electron detector> uses gas amplification to collect secondary-electron signals under these conditions. Gas <pressure>, path length and scattering must be controlled because they also affect the probe and background.<Low-vacuum SEM by JEOL>[glossary entry: gas-ion charge neutralization and environmental secondary-electron detection]{ref}
Finally, acquire an <energy-dispersive X-ray spectroscopy> spectrum at each pixel to add elemental specificity. The beam creates inner-shell vacancies, and characteristic X-rays emitted as they fill identify chemical elements. Selecting fitted characteristic-line intensities yields elemental maps that can be compared with the <electron> image. Peak overlap, detector response, absorption and the X-ray-generating interaction <volume> limit identification, quantification and spatial resolution. This gives chemical-element contrast; resolving different bonds or oxidation states generally needs more detailed spectral information than one broad elemental map.
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