Although light's wavelength is too large to examine objects much smaller than a biological cell, one method to explore structures on the atomic scale (and smaller) is to toss particles with even smaller wavelengths at those structures, and see what happens if and when they collide. The basic equation for tracking interactions like these is a range equation, very much like that used for radioactive decay or even the compounding of interest. Instead of a "half-range for interaction" analogous to "half-life for radioactive decay", it's customary to use an "e-folding length" or "mean-free-path" τ for interaction. This is basically the distance traversed before 1/e of the incident beam has undergone the interaction of interest. If I_o is the incident intensity and I the non-interacted intensity after traversing a thickness t, this relationship is written: I = I_oe^-t/τ. Here "e" is the base of the natural logarithms, and the only non-zero number whose x^th power equals its own slope. That's why the expression for I above describes a physical situation where the rate of non-interacted beam loss dI/dt is proportional to the amount of non-interacted beam I, i.e. dI/dt = -I/τ.

The dots denote cross-sections for a variety of specific interactions. The grey dots are for inelastic scattering of a 300 keV electron passing through a specimen of solid Silicon, as well as for an electron near its most strongly interacting energy in a metal: around 70 eV. Because electrons interact so strongly with matter, the mean free paths are quite small. At 300 keV, heavier atoms (e.g. a gold specimen) will have approximately the same interaction area per nucleon but an even shorter distance mean-free-path (as indicated by the gold dot). These dots might be replaced with curves parameterized according to electron energy in the days ahead. You might already imagine that as electron energy increases beyond 70 eV, the cross-section for inelastic scattering goes down. This could prove important, for example, in deciding what electron energy to use if your specimen is beam-sensitive.

Another classic example of cross-section logic involves a technique now called "Rutherford backscattering". It provided our first compositional data beamed back from a rocket impacting the moon, and is presently used to examine the depth profile of heavy atoms in semiconductor specimens. However it was originally used by Ernst Rutherford to argue that the large amount of backscatter could only be explained by a concentration of repulsive charge at the center of those gold atoms (now referred to as the atomic nucleus) . His model considered only an unscreened repulsive nucleus, but nonetheless worked well for large scattering angles. In an experiment bombarding a gold foil with 5 MeV alpha particles, where would the cross-section for scattering that doubly-charged ion backwards (>90 degrees) plot on the graph above? For example, would the answer be near to that faint yellow dot on the graph? [Hint] Applications that worry about cross-sections for scattering through one angle, or another, include astrophysical studies of gravitational lensing, and a wide range of diffraction and imaging experiments with electrons, neutrons, ions, and light.

Neutrons, being uncharged, interact much more weakly with matter. The black dot denotes one of the highest cross-section neutron interactions, namely that for absorption of a thermal neutron by U₂₃₅. As you can see, the mean free path is much larger than for electrons in solids: about one third of a millimeter. The effective cross-section per atom is hundreds of times the geometric area of the nucleus. Ordinary atomic cross-sections for neutron interaction are typically closer in size to the geometric cross-section of the nucleus (about a barn).

Mean free paths are also relevant to thermal transport in gases, liquids and solids. For example, the mean free path between collisions of nitrogen molecules in air at standard temperature and pressure is suprisingly short, as shown by the darker blue-green dot at the right of the figure. This suggests that air is actually rather crowded, in spite of the fact that our perceptions are very finely tuned to ignore it (and thus to in effect take it for granted).

In addition to parameterizing curves according to electron, neutron and atom energy, we also plan to add curves for interaction of various targets with electromagnetic radiation in days ahead. For example, the brightness of an object (like a flat panel display) in a "pea-soup fog" might fall off by a factor 1/e~1/2.7 for each added meter of fog between you and it. The bright cyan dot in the figure above shows approximately where the mean free path for light scattering by that fog might plot on our interdisciplinary map of cross-sections.

For a question easier to answer in the lab, we might ask: What is the fraction of the light absorbed by a 10nm thick carbon support film? Can you design an experiment to check this? The figure below might give you some ideas. (P.S. While you're pondering that figure, can you guess how all forty of the 4x6 square-arrays on that grid are indexed with a tiny carved symbol from A-Z or 1-14?) Given the fraction of light absorbed, could you then calculate a mean free path and thus add a dot to the graph above? What other targets might be fun to consider in this way? Note: Cross-sections for neutrino interaction with matter might be too small to make it onto the plot, unless we expand axes down and to the left another 20 orders of magnitude. The number of grams per square centimeter of shielding needed to shield you from an X-ray beam would lie somewhere in between.

Contrast in microscopy generally involves cross-sections for one sort of scattering or another. For example, it might involve cross-sections for Bragg or diffuse scattering through a given angle, or cross-sections for loss of a certain amount of energy, or cross-sections for reflection or for absorption. We illustrate with a couple of recent examples that we've been thinking about below.

One of the easiest macroscopic properties of a substance to measure on earth is its weight, and by inference therefore the amount of mass contained therein. Since inelastic scattering from a substance is often tied to the number of charged particles it contains, measurements of inelastic scattering cross-section (which for high energy electrons in grams/cm² are comparable for many different target atoms) can provide local microscopic estimates of proton number and by inference mass down to the atomic scale, for comparison to other microscopic quantities (e.g. number or surface area of particles).

The figure below right, taken by Eric Mandell with Howard Berg's energy filtered transmission electron microscope (TEM) at the Donald P. Danforth Plant Sciences Center, shows a mean-free-path "thickness image" profiled (below left) against the expected thickness profile (blue curve) of a bamboo-style multiwall carbon nanotube. The thickness image (following the range equation above) is the logarithm of the ratio between an unfiltered brightfield image of the tube (which shows all electrons not scattered outside of the optical path, or Io[x,y]), and a "zero-loss" image (which shows only those electrons I[x,y] which did not lose appreciable energy). As you can see, the physical thickness model agrees quite well with the measured thickness model, even though the varying orientation of the graphite layers in the nanotube wall result in very different amounts of "scattering outside of the optical path", in this case largely due to electron diffraction. The red curve profiles intensity diffracted by graphite (002) planes for comparison.

Unlike the case of a complete unknown, an independent estimate of the mass of a carbon nanotube can be obtained by taking advantage of its approximate cylindrical symmetry, and our knowledge of the density of graphite (around 2.2 g/cc). For example from the information given about the figure above (including the scale bar in the image), can you estimate the total mass of tube material covered by the yellow rectangle?

This image, taken with equipment in the MIST lab at UM-StL with specimens developed by Lifeng Dong and others at Missouri State University in Springfield, illustrates weak scattering of 300 kV electrons by single DNA strands as well as the obscuring effect of a 10nm thick carbon support film (the curved material at the right of this image). Although we are imaging the electron wave field near to (but downstream of) the specimen, intensity in thin specimens like this is essentially a convolution of the specimen's cross-section for interaction (proportional to the projected potential) with the microscope's point spread function. As you can see, the fractional intensity loss due to a single phosphate backbone is quite weak even compared to the intensity loss through a "quite transparent" 10nm carbon film. It is possible that the film that recorded this image contains quantitative information on the scattering cross-section for 300 kV electrons by DNA strands, the carbon support film, and the "known thickness" single walled carbon nanotubes? If so, might you be able to determine the number of protons per cm² that each of these objects are putting "in the way of the beam"?

There is also information in lateral intensity variations. The familiar 3.5Å separation between van der Waals bonded (002) layers of graphite can be seen in the projected spacing between walls of those nanotubes which have more than one wall. Edge-on layers of Pt atoms with their characteristic 2.3Å (111) separation can also be seen in those tiny crystals of Platinum. A double-helix on this same size scale would be as wide as the 2nm scale bar at bottom left. The average length of a human chromosome double-helix un-supercoiled, on the same scale, would be about 60 miles.

Would you like to explore the the raw data that provided the image above? How about electron phase contrast images of some stuff that spontaneously combusts, a single crystal with 1.92Å cross-fringes and strange inclusions, a flame-manufactured carbon nanotube only 11Å in diameter, where it landed after being cut by the electron beam, and some sectioned dissolution residue from the Murchison meteorite with sliced interstellar graphite onions. What kinds of quantitative data can you find in them?

Aberration-corrected high angle annular darkfield (HAADF) images, taken in collaboration with colleagues at Oak Ridge National Lab, allow one to easily see heavy atoms in a matrix of much lighter atoms as shown in the figure below. If the specimen in the image below is a wedge of carbon (density about 2 g/cc) with thickness running from 0 at the top to 10 nm at the bottom of the field of view, how many heavy atoms per cubic centimeter do you see evidence for in this specimen? The amount of scattering from each cluster of heavy atoms (as discussed above) may someday allow us to determine the total atomic number of each as well.

Note that the red-green channels are the HAADF image, while the blue channel contains the brightfield (unscattered beam) signal recorded at the same time. This is more sensitive to the several 3.4Å graphite (002) fringes seen edge-on running from left to right along the top edge of the specimen. The rest of the specimen is comprised of unlayered graphene, a novel form of carbon found so far only in the core of presolar graphite spheres condensed in the asymptotic giant branch stars which manufactured most of the carbon atoms on earth. Its scattering signature is easiest to see in diffraction patterns, as well as in the power spectrum of this image.

A closeup of one of those heavy atoms in the thinnest (top) part of the specimen shows that scattering from that atom comes from a point much smaller than the typical 2Å spacing between atoms. In fact, the odd shape of the "center spot" is largely a result of residual aberrations affecting the shape of the sub-Angstrom beam spot. The same thing happens in SPM (tunneling and atomic force) images when the specimen is sharper than the tip: The specimen images the probe rather than vice versa!

To illustrate how quantitative data on such scattering processes might come in handy downstream, in the figure below we've measured the integrated intensity under a series of "mostly single-atom" peaks from the image above, and are comparing the resulting intensity histogram to a plot of astrophysically observed stellar photosphere abundances. Although the "Z-scale" for the experimental histogram has not yet been calibrated, it's easy to see how such images might "fingerprint" the heavy elements (many of them manufactured in that same star's interior) that are present in the specimen, as well as how those atoms are distributed within the grain.

As a first step toward calibrating the "Z-scale", below are a couple of HAADF image simulations calculated by Eric Mandell at UM-StL using modified versions of Kirkland's C algorithms. The "specimens" are 128Å square films 40Å thick of amorphous-carbon, wedge-thinned to perforation along a horizontal line in the center with a small number of Fe or Zr atoms placed randomly therein. What atomic numbers do you think are associated with the atoms in the experimental image?

Some experimental clues to specimen thickness in protons per cm² (the high angle scattering goes roughly as Z^1.7) as one moves perpendicular to the specimen edge in the experimental images are shown by the ImageJ profiles below. These average intensity across columns 202 through 276 in the image which started this section. As in that image, the brightfield intensity is in blue, while the annular darkfield intensity is in yellow.

In this note, we've avoided angle-specific scattering processes like diffraction, Kikuchi-line formation, strain contrast, and lattice fringe imaging. One reason is that they are highly dependent on specimen symmetry and orientation, a subject more appropriate for a materials microscopy module than for the generic module of which this note is a part. Hints about the coolness of that stuff include the animations at the top of this page: The animation at top left takes you on a trip through the reciprocal lattice of a buckyball with red (110) and blue (100) graphene shells as benchmarks, while at top right you get to visit all four 3-fold zones on the lattice fringe visibility map of a face-centered cubic crystal. Some clues to recent applications and developments in these areas are linked below.

• A mean-free-path analysis protocol for STEM image
• A simple electron diffraction simulator
• Notes on lattice exploration in 3D.
• Notes on atom clusters with icosahedral twinning.
• A draft paper on harmonic analysis techniques for Microscopy and Microanalysis 2007.