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I am asked to determine the resolving power of both a TEM (transmission electron microscope) and a SEM (scanning electron microscope) and given the Rayleigh formula below.

$$Resolution=\frac{0.61\lambda}{N.A.}$$ Where $\lambda=0.0039$nm since the wavelength of an electron is $0.039$ angstroms and $1Å=0.1$nm.

I am also told that the N.A. the numerical aperture for the TEM is $0.01$

So, $$Resolution=\frac{0.61(0.0039nm)}{0.01}=0.24nm$$

For the SEM I am not given the N.A. but I am told that the resolution is $10$nm

My problem is if I use the Rayleigh formula $$Resolution=\frac{0.61\lambda}{N.A.}=10nm$$ I get $$N.A.=0.0002379$$ Which would imply that the aperture for the SEM is greater than that of TEM since N.A for TEM is $0.01$ and SEM is $0.00023$ $$0.01>0.00023$$.

My problem is that I'm told the greater the aperture (smaller value of aperture) the greater the resolution but as you can see from the equations above this doesn't seem to make sense, The only explanation I can think of is that the Rayleigh formula does not apply to electron microscopes. Any advice would be greatly appreciated thank you.

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This is more than likely better asked in physics SE, however since you contextualized it biologically somewhat I will answer it here.

You are correct in your concern. The Rayleigh Criterion and the associated calculations(Rayleigh Scattering) are talking about the minimum separation of 2 light sources to resolve a given object. From this we are able to derive nice metrics like resolution and so forth. As you say, this describes the resolving of light wrt to the diffraction limit.

In order to calculate the resolving power of an SEM or TEM you need to do a different set of calculations.

First, the theoretical limit of EM microscope resolution is given by the De Broglie Wavelength of the accelerated electrons. However, as with the calculations for the Rayleigh Criterion, there are other factors to consider. For a full walkthrough see this. You take Abbe's equation for resolution in a perfect optical system, and take account of the other factors at play(optics for example).

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The resolution of an SEM is not set by the electron wavelength, but rather by the size of the electron beam that is scanned over the sample. Imagine you had a flashlight scanning over a patch of ground, and were recording the reflected light intensity, without using any lenses, to reconstruct the surface features -- your resolution would be roughly the spot size, and not the Rayleigh limit. It's important to keep in mind that the Rayleigh limit is the best resolution one can have; it's easy to do worse!

A TEM, in contrast, does use lensing of electron beams, via magnetic fields.

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