# I Are the electrons in an electron microscope coherent?

1. Feb 13, 2018

### gnnmartin

In an old fashioned electron microscope (the type I was meant to understand at university 50 years ago), are the electrons coherent, or do we just consider an electron interfering with itself? If they are coherent, how are they made coherent?

2. Feb 13, 2018

### ZapperZ

Staff Emeritus
I do not understand what you mean by "coherent" in this context. If you mean that they share the same wavefunction with a coherent phase, then the answer is no.

Even the electrons in a SEM device are produced from a thermionic cathode. You do not get "coherent" electrons that way. Besides, in a standard electron microscope, there isn't any need for a coherent source. You are not measuring "interference" effects or any effects that require such properties.

Zz.

3. Feb 13, 2018

### sophiecentaur

Not "coherent" but I believe they use a narrow band of energies, which means the equivalent of monochromatic. For best resolution, the de Broglie wavelength would be a short as possible (requiring high energy electrons)
Also, the beam has to be 'Colimated' - another C word.

4. Feb 13, 2018

### Andy Resnick

That's an interesting question- never thought of it before.

From my limited understanding, the electron source (gun) has a small cross-section area to approximate a point source, and as mentioned above, they are also 'monochromatic' to some extent, indicating that there is some degree of coherence in the illumination beam. However, the corresponding wavelengths are so much smaller than for visible light it's hard to be specific.

Quantitatively, the coherence area A and coherence length 'L' are readily defined: L = 2π v/Δω, where v is the velocity and Δω the frequency spread of the beam; this is likely related to the voltage spread at the source. A =λ2/ΔΩ, where λ is the mean wavelength and ΔΩ the apparent size of the source.

These expressions are easy to work with for light (v = c, simple relationship between ω and λ), less so for electron beams. Try working it out!

5. Feb 13, 2018

### ZapperZ

Staff Emeritus
But these "coherent" quantities have more to do with the physical dimensions of the illuminated area when compared to the deBroglie wavelength of the electrons. It doesn't mean that the electron beam itself is coherent, not in the sense that a laser beam is coherent. This is why I asked the OP on what is meant by "coherent" in his/her post.

Coincidentally, my avatar is an electron microscope image (SEM image) that I took of a "melted" region of copper that had undergone an electrical breakdown.

Zz.

6. Feb 14, 2018

### gnnmartin

Thanks for your replies, and they have helped me reduce my confusion. I was trying to understand why an electron (or a photon) only seems to interfere with itself. As Zapper says, collimation & narrow energy band ensure that the probability distribution of each electron considered individually match, summing to form the expected interference pattern.

7. Feb 14, 2018

### ZapperZ

Staff Emeritus
What interference pattern?

And I didn’t say that. It was from @sophiecentaur .

Zz.

8. Feb 14, 2018

### Andy Resnick

"Coherence" is indeed a property of the illuminating beam and it does impact many aspects of imaging systems. The question is interesting (to me) because I have never seen the coherence length or volume calculated for an electron microscope beam.

I forgot, there's also a spin degree of freedom...

http://aip.scitation.org/doi/full/10.1063/1.4901745
https://www.ncbi.nlm.nih.gov/pubmed/23545433

9. Feb 14, 2018

### gnnmartin

Back in the early 60s I heard a lecture or two about using electron microscopes to examine the structure of a crystal. It was explained in terms of interference patterns, in much the same way as light passing through a grid forms patterns that can be explained as interference patterns. It still seems a reasonable way to explain the patterns, though I admit that I find it hard to extend the explanation to the image you use as an avatar.

10. Feb 14, 2018

### ZapperZ

Staff Emeritus
But that's not what a typical "electron microscope" does, which is the topic of your question. A standard electron microscope does imaging, not spectroscopy.

Now, an electron microscope, especially nowadays, may be fitted with the capability to do RHEED or LEED, i.e. to look at diffraction patterns, etc. to study the crystal lattice, but this is not what a normal electron microscope does.

Zz.

11. Feb 14, 2018

### sophiecentaur

If you send a beam of electrons through a thin graphite film, for instance, there is a resulting diffraction / interference pattern (a few concentric rings). The beam is only 'coherent' to a very low degree, in that it is narrow and travelling in one direction. The pattern is of similar quality as the optical interference patterns you get from monochromatic or narrow band filtered beam of light. The pattern is the statistical sum of the probabilities of individual electrons of photons arriving at a particular place on a receiving 'screen'. The "interfering with itself" interpretation is valid because the arrival point may not be in line with any straight path through the system. The particles, arriving one at a time and with no chance to 'interfere with each other', will still form an interference pattern so they must be behaving like waves on the journey through the structure. The more coherent the beam is, the better quality is the pattern. Easy to achieve with a photons from a laser but harder with a beam of electrons. I am not sure what the limit would be for this. I don't know of any methods, other than using a tight velocity selector and a very small slit?

12. Feb 14, 2018

### ZapperZ

Staff Emeritus
But you use this in getting a real image in an electron microscope? I find that hard to believe!

Zz.

13. Feb 14, 2018

### sophiecentaur

Of course it's a real image. Young's slits patterns are real images too. The electrons build up in some places and not in others. The 'strongest' pattern is a shadow of the test object and the fringy bits are the same as the fuzz round the features you can see with an optical microscope at high mag. (Or isn't that what you meant?)

14. Feb 14, 2018

### ZapperZ

Staff Emeritus
No, I mean "real" as opposed to "reciprocal". Diffraction images are images representing reciprocal space.

SEM and electron microscopes are typically used to look at the real space images. How do you use diffraction/interference pattern to see what I am seeing below?

Zz.

15. Feb 14, 2018

### Andy Resnick

The first paper I linked to calculated the transverse coherence length of their system as " (1.7 ± 0.3) × 102 nm", which would seem to indicate speckle patterns beginning to occur when imaging features 100 nm or smaller.

What's more interesting (to me) is that transverse coherence length seems to be easier to measure than longitudinal coherence length (monochromaticity); I would expect 'chromatic aberration' to be a critical metric when designing the magnetic lenses.

16. Feb 14, 2018

### sophiecentaur

What you see is always a diffraction pattern. It's just that, when you are well within the 'resolution' limits of the system, the errors due to diffraction effects are negligible.
The diffraction images that you get from Xrays when they pass through a crystal are significant because the wavelength is of the order of the lattice spacing. There is still a major image on which most of the beam falls and that is in the straight through direction - a slightly diffuse dark spot. You get that for any size of lattice or its orientation. It depends on what you want to find out about a structure whether you try to utilise or minimise the effects of diffraction.This is the same as the image of a star that you get at low magnification but, as you increase magnification, you get the defects in the optics, some of which are due to diffraction.
I suppose that would depend upon the 'bandwidth' of the electron energies that are selected in the electron gun - traded off against the obtainable beam power and the associated signal to noise ratio.

17. Feb 14, 2018

### ZapperZ

Staff Emeritus
I really do not understand any of this in terms of how one uses an electron microscope. You do NOT want any diffraction effects, because it will blur your images! If this were to occur in a TEM, the image will be garbage.

The image that I showed in my post is from a SEM device. It means that it is a collection of secondary electrons from the material that is being looked at. This means that it is not even the same electron that came in from the cathode!

So, once again, what "interference" and "diffraction" effects are necessary in doing real space imaging of the material in an electron microscope? And please keep in mind that I am quite familiar with XRD, LEED, and RHEED techniques.

Zz.

18. Feb 15, 2018

### Andy Resnick

That's an essential difference- I wouldn't expect the secondary electrons to have the same degree of coherence as the illuminating beam due to inelastic scattering.

19. Feb 15, 2018

### Andy Resnick

No- coherent effects can be exploited in imaging: phase contrast is the obvious method, and it is being developed in TEM systems:

http://rstb.royalsocietypublishing.org/content/363/1500/2153

but not the only one- there are also analogies to DIC imaging in TEM machines.