λ =
h/p = h/mv
λ :
wavelength; h: Planck constant; p: momentum
The energy
of accelerated
electrons is equal to their kinetic energy:
E = eU
= m_{0}v^{2}/2
U: acceleration voltage
e / m_{0} /
v: charge / rest mass / velocity of the electron
These
equations can be combined to calculate the wave length of
an electron
with a certain energy:
p = m_{0}v
= (2m_{0}eU)^{1/2}
λ =
h / (2m_{0}eU)^{1/2} (≈ 1.22 / U^{1/2} nm)
At the
acceleration voltages used in TEM, relativistic effects have
to be taken into account (s. Table):
λ =
h / [2m_{0}eU (1 + eU/2m_{0}c^{2})]^{1/2}

Relativistic
wavelength /
pm

Mass
x m_{0} 
Velocity
x 10^{8} m/s 
100 
3.70 
1.20 
1.64 
200 
2.51 
1.39 
2.09 
300 
1.97 
1.59 
2.33 
400 
1.64 
1.78 
2.48 
1000 
0.87 
2.96 
2.82 
Rest mass of an electron: m_{0} = 9.109 x 10^{31} kg
Speed of light in vacuum: c = 2.998 x 10^{8} m/s