Finding diffraction-limited spatial resolution

[leroyjenkens]

Homework Statement

A scanning electron microscope (SEM) images samples using a beam of electrons. If
the electrons are accelerated across a potential difference of 5.00 kV, what is the
diffraction-limited spatial resolution of the SEM? Work this problem both relativistically and nonrelativistically.

Homework Equations

This is what I used, but I'm not sure what exactly to use.
$$E=eV$$
$$E=\frac{hc}{λ}$$
$$E=\frac{mv^{2}}{2}$$

The Attempt at a Solution

I first solved for E using E=eV, because I have both e and V, then I plugged that into the second equation to solve for wavelength (don't know why, I just wasn't sure what I'm supposed to be solving for. I understand what the question wants from me).

Then I set $eV=\frac{mv^{2}}{2}$, solving for velocity and found it to be 4.19x10⁷.

Then I solved for E using $E=γmc^{2}$

But I'm not sure what the question wants me to find. What is diffraction-limited spatial resolution?

Thanks

 

[mark726]

for your question! your main goal is to understand and explain natural phenomena through empirical evidence and logical reasoning. In this case, the question is asking for the diffraction-limited spatial resolution of the SEM, which is the smallest distance between two points that can still be distinguished by the microscope's imaging system. This is a measure of the microscope's resolving power, or its ability to distinguish between small details in a sample.

To find the diffraction-limited spatial resolution, you can use the de Broglie wavelength equation, which relates the wavelength of a particle to its momentum and mass. In this case, the particles are electrons and their momentum can be calculated using the relativistic equation you mentioned, E=γmc^2. This equation takes into account the relativistic effects at high velocities, which are important for electrons accelerated to 5.00 kV.

So, to solve for the diffraction-limited spatial resolution, you can follow these steps:

1. Calculate the energy of the electrons using E=eV, where e is the charge of an electron and V is the potential difference of 5.00 kV.

2. Use the relativistic equation E=γmc^2 to find the momentum of the electrons.

3. Substitute the momentum into the de Broglie wavelength equation, λ=h/p, where h is Planck's constant and p is the momentum.

4. This will give you the de Broglie wavelength, which is the diffraction-limited spatial resolution of the SEM.

5. Repeat the calculation using the nonrelativistic equation, E=mv^2/2, and compare the results to see the difference between the relativistic and nonrelativistic calculations.

I hope this helps you in your solution! Remember to always think critically and consider the physical meaning of the equations you are using. Good luck!