Properties of Electrons

The dualism wave/particle is quantified by the De Broglie equation:

λ = h/p = h/mv

λ : wavelength; h: Planck constant; p: momentum

The energy of accelerated electrons is equal to their kinetic energy:

E = eV = m₀v²/2

V: acceleration voltage
e / m₀ / 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 = m₀v = (2m₀eV)^1/2

λ = h / (2m₀eV)^1/2 (≈ 1.22 / V^1/2 nm)

At the acceleration voltages used in TEM, relativistic effects have to be taken into account (s. Table):

λ = h / [2m₀eV (1 + eV/2m₀/c²)]^1/2

 

Vacc / kV	Non relativistic wavelength / pm	Relativistic wavelength / pm	Mass x m0	Velocity x 108 m/s
100	3.86	3.70	1.20	1.64
200	2.73	2.51	1.39	2.09
300	2.23	1.97	1.59	2.33
400	1.93	1.64	1.78	2.48
1000	1.22	0.87	2.96	2.82

Rest mass of an electron: m₀ = 9.109 x 10^-31 kg
Speed of light in vacuum: c = 2.998 x 10⁸ m/s