deBroglie's electrons...

Electron Wave-Bug & Animation Link

... & some remarkable TEM facts.

This is a note to an electron microscope (TEM stands for Transmission Electron Microscope) listserver group which will eventually be put on a page of its own. It discusses the practical example of relativistic acceleration in an electron microscope, like that shown below:

Our Philips 300kV SuperTwin Electron Microscope

What follows is a list of some physical perspectives on the electrons used routinely in transmission electron microscopy. Without knowing it, you may on a daily basis be putting to use things, like the wave nature of electrons, that were inconceivable in the early part of this century. In fact, some of the properties of these electrons may be only marginally conceivable today! For example, note from the animation in the upper right corner of this page that the (complex-color) phase velocity of the electron is greater than the group velocity of the packet itself. What's even more bizarre (I think) is that the phase velocity is always faster than the speed of light! But, of course, that's only the beginning...

Other resources of possible interest:
  • Quantum mechanics links from the Math Institute at Oxford.
  • Animate electron and photon wave packets. What's different?
  • Try focussing a high-res electron microscope image on-line!
  • How are Balinese candle-dancing and electron spins connected?
  • Cite/Link:
  • This release dated 23 Oct 1996 (Copyright by Phil Fraundorf 1988-1996)
  • This note has later published in a microscopy newsletter (was it Microscopy Today?)
  • The original Jan 15 1996 list-server post is still available (5/2005) here.
  • Since 22 Feb 1997, you are visitor number [broken counter].
  • Mindquilts site page requests ~2000/day hence approach a million per year.
  • Requests for a "stat-counter linked subset of pages" since 7 April 2005: .
  • Notation Key is: coordinate {time, velocity}={t, v}, and proper {time, velocity}={T, w}.
  • Start relativity with the metric equation instead of Lorentz transforms!.
  • Does making a hotdog require 50 nanoseconds of life's power stream?
  • Figure traveler time on a "1-gee" trip to Andromeda galaxy.
  • Browser-interactively solve your own constant-acceleration problems.
  • Is statistical physics dead, or is there a paradigm change afoot?
  • At UM-StLouis see also: cme, i-fzx, phys&astr, programs, stei-lab, & wuzzlers.
  • Some current and previous courses: p111, p112, p231, p341, p400.

    Fast electrons: a back of the envelope calculation for 300 keV electrons gives γ = (300+511)/511 = 1.587, so that they travel at v = c Sqrt[1-(1/γ)^2] = 0.777 c or [lightyears per coordinate year] of elapsed time. However, if we consider traveler (i.e. electron or proper) time for such a speeding electron, this would give that they travel w = γv = 1.232 [lightyears per traveler year] of elapsed time! With this spatial 4-vector (or proper) velocity well over c, we're dealing with relativity in action! I wonder how many g's of acceleration they experience in the electron gun in order to get up to speed? For more on this subject, you might want to check our browser-interactive relativistic Accel-One problem solver, and the theory pages attached.

    Lonely electrons: I think it was John Armstrong of Caltech who once pointed out to me that the number of microscope beam electrons in your TEM specimen at any one time is so small that the odds of such electrons interfering with each other to form diffraction patterns is quite small. The vertical separation between electrons in the column is ve/I where I is the specimen current, v is the electron coordinate-velocity, and e is the charge per electron. For a nanoamp of 300 kV electrons, this is (0.777×3×108m/s)×(1.6x10-19C/e-) / (10-9C/s) = 0.037 meters/e-. Under some illumination conditions there may be no more than 1 beam electron in the column at a time! Hence diffraction patterns in the TEM are basically formed by individual electrons interfering with themselves! As you know, such interference will occur only if we don't take steps to determine the path of individual electrons through the specimen! If we look too closely at these paths, the diffraction patterns would disappear (cf. Englert et al., Scientific American, Dec. 1994, 86-92 on quantum erasure).

    Fat electrons: The transverse coherence widths of electrons which make possible electron phase contrast (HREM) lattice imaging and probably electron holography might also be seen as lateral broadening of individual electron wave-packets via the uncertainty principle, which results because we know too much about their transverse momentum! For example, if Δy is the transverse width and Δpy is the transverse momentum spread equal to angular spread alpha times momentum mw, the uncertainty principle gives Δy Δpy >= h/4π or Δy >= h/(4π alpha mw). For alpha=0.2 milliradian with 300 kV electrons, this gives lateral wave-function spreads of about 15 Angstroms in a LaB6 HREM, while field emission sources subtending a smaller angle at the specimen may yield widths larger than 100 Angstroms. Are these numbers reasonable? By increasing the spread of electron angles in the incident beam, this transverse coherence width can presumably be decreased (e.g. if you want it small for Z-contrast imaging), or varied as in the variable coherence-width strategies of Murray Gibson at U. of I.

    Long electrons: The tight tolerences on high voltage stability and the emitted spread in electron energies means that our uncertainty in the longitudinal momentum of TEM electrons is quite small, and hence again by the uncertainty principle that the wave-packet spread in the direction of motion for TEM electrons can be quite large. For example, if Δx is the longitudinal width and Δpx is the longitundinal momentum spread equal to ΔK/v where ΔK is the energy spread, then the uncertainty principle gives Δx Δpx >= h/4π or Δx = vh/ΔK. For K=300kV electrons with a 0.6 volt energy spread ΔK, this is more than 1000 Angstroms! The associated tight distribution of incident electron energies decreases chromatic and instability damping of fine details in CTEM and HREM images, so that for most applications you may want your electrons "as long as possible". An exception might be in variable-coherence strategies (mentioned above), where shorter electrons might provide sensitivity to shorter-range vertical correlations.

    Focussed electrons: Briefly I'll mention only that the focussed probe in a scanning electron microscope is great for imaging "large objects" like bugs because the depth over which the electron beam remains focussed is large enough to let the top and bottom of the insect be sharply imaged at the same time. Conventional light microscope images of bugs, by comparison, often have the antennae out of focus when the feet are in focus, so that scanning electron microscopes win the prize hands down for the "scariest bug images". All of this is nothing, however, to the claim that high magnification transmission electron microscope images remain in focus not just on the viewing screen and on the film a few centimeters below, but in fact might be in focus even if the film were displace downward by a kilometer for more! Does anyone want to verify, and explain why this is so?

    The foregoing thoughts on fast, lonely, fat, long and focussed electrons are not really things I've had time to think much about, but they are interesting, and hence I would enjoy other perspectives on them as well as suggestions for other "remarkable TEM facts" to add to the list!

    Note: Send comments and questions to This page might contain original material. Hence if you echo, in print or on the web, a citation would be cool. {Thanks. :) /pf}