Question

(a) Calculate the wavelength of an electron under an acceleration voltage of (i) 200 V, (ii) 1.5 MeV and (ii) 2.5 GeV. Also, calculate the velocity of electrons moving at these acceleration voltages. How does this velocity compare to the speed of light. If they are comparable to the speed of light, use relativistic effects to calculate the wavelength of electron?

(b) If the wavelengths are so small, can one expect the resolution of the TEM made with these energies to be so small? If not, what will be the reason behind those discrepancies.

Answer 1

Velocity of the electron is determined by the accelration voltage or electronpotential, which is given by:

where V= accelratin potential,

v= velocity of electron

e= charge on an electron

m= mass of electron

Now velocity calculation is:

i.) At 200 V

  

  m/s

Velocity of light is App.  m/s

Comment: Hence the velocity of electron is 0.02796 times slower then velocity of light.

ii.) At 1.5MeV

  

  

  m/s

Velocity of light is App.  m/s

Comment: Hence the velocity of electron is 2.417 times faster then velocity of light.

iii.) At a potential of 2.5GeV

  

  

  m/s

Velocity of light is App.  m/s

Comment: Hence the velocity of electron is 2981.33 times faster then velocity of light.

Now calculating wavelength of of the electron in all 3 conditions.

Wavelength is given by:

where V= accelratin potential,

v= velocity of electron

e= charge on an electron

m= mass of electron

h= constant

or after simplification of constants, we got

i.) At 200 V

Hence the wavelength is about an Angestrom. so we can expect the resolution of the TEM with these small waveength.

ii.) At 1.5MeV

Hence the wavelength is about an Picometer. so we can't expect the resolution of the TEM with these small waveength.

iii.) At a potential of 2.5GeV

Hence the wavelength is very small so we can't expect the resolution of the TEM with these small waveength.