Difraccion de electrones

LD
Physics
Leaflets

Atomic and Nuclear Physics
Introductory experiments
Dualism of wave and particle

P6.1.5.1

Diffraction of electrons in
a polycrystalline lattice
(Debye-Scherrer diffraction)

Objects of the experiment
g Determination of wavelength of the electrons
g Verification of the de Broglie’s equation
g Determination of lattice plane spacings of graphite

PrinciplesLouis de Broglie suggested in 1924 that particles could have
wave properties in addition to their familiar particle properties.
He hypothesized that the wavelength of the particle is inversely proportional to its momentum:
λ=

h
p

(I)

λ: wavelength

D1

h: Planck’s constant
p: momentum
His conjecture was confirmed by the experiments of Clinton
Davisson and Lester Germer on thediffraction of electrons at
crystalline Nickel structures in 1927.
In the present experiment the wave character of electrons is
demonstrated by their diffraction at a polycrystalline graphite
lattice (Debye-Scherrer diffraction). In contrast to the experiment of Davisson and Germer where electron diffraction is
observed in reflection this setup uses a transmission diffraction type similar tothe one used by G.P. Thomson in 1928.

Bi 0206

D2
Fig. 1: Schematic representation of the observed ring pattern due to
the diffraction of electrons on graphite. Two rings with diameters D1 and D2 are observed corresponding to the lattice
plane spacings d1 and d2 (Fig. 3).

From the electrons emitted by the hot cathode a small beam
is singled out through a pin diagram. After passingthrough a
focusing electron-optical system the electrons are incident as
sharply limited monochromatic beam on a polycrystalline
graphite foil. The atoms of the graphite can be regarded as a
space lattice which acts as a diffracting grating for the electrons. On the fluorescent screen appears a diffraction pattern
of two concentric rings which are centred around the indiffracted electron beam(Fig. 1). The diameter of the concentric
rings changes with the wavelength λ and thus with the accelerating voltage U as can be seen by the following considerations:

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P6.1.5.1

LD Physics leaflets

-2-

From energy equation for the electrons accelerated by the
voltage U
e ⋅U =

1
p2
m ⋅ v2 =
2
2⋅m

n λ = 2d sinϑ

(II)

U: accelerating voltage
e: electron charge

ϑ

m: mass of the particle

ϑ

v: velocity of the particle

∆1

the momentum p can be derived as
p = m ⋅ v = 2 ⋅ e ⋅m ⋅U

∆2

d

(III)

Substitutingequation (III) in equation (I) gives for the wavelength:
λ=

h

(IV)

2 ⋅ m ⋅ e ⋅U

In 1913, H. W. and W. L. Bragg realized that the regular arrangement of atoms in a single crystal can be understood as
an array of lattice elements on parallel lattice planes. When
we expose such a crystal lattice to monochromatic x-rays or
mono-energetic electrons, and, additionally assuming that
thosehave a wave nature, then each element in a lattice
plane acts as a “scattering point”, at which a spherical wavelet forms. According to Huygens’ principle, these spherical
wavelets are superposed to create a “reflected” wave front. In
this model, the wavelength λ remains unchanged with respect
to the “incident” wave front, and the radiation directions which
are perpendicular to the two wavefronts fulfil the condition
“angle of incidence = angle of reflection”.
Constructive interference arises in the neighbouring rays
reflected at the individual lattice planes when their path differences ∆ = ∆1 + ∆2 = 2⋅d⋅sinϑ are integer multiples of the
wavelength λ (Fig. 2):
2 ⋅ d ⋅ sin ϑ = n ⋅ λ

n = 1, 2, 3, …

Fig. 2: Schematic representation of the Bragg condition.

If we...