Page 44 - Electrical Properties of Materials

Introduction 27

Detector
Electron gun

Reflected electrons
Incident electron Fig. 2.3
beam Schematic representation of Davisson
d sinθ and Germer’s experiment with low
θ θ energy electrons. The electrons are
• • • • • effectively reflected by the surface
d layer of the crystal. The detector
• • • • • shows maximum intensity when the
d individual reflections add in phase.

the electrons should have wavelike properties with a wavelength inversely
∗
proportional to particle momentum, namely h is Planck’s constant (not the
height of the waves in the ripple
h
λ = . (2.6) tank) with a rather small numerical
mv –34
value, namely 6.6 × 10 J s, and
To test de Broglie’s hypothesis, Davisson and Germer fired a narrow beam m and v are the mass and speed of
of electrons at the surface of a single crystal of nickel (Fig. 2.3). The wave- the electron.
like nature of the electron was conclusively demonstrated. The reflected beam
displayed an interference pattern.
The arrangement is analogous to a reflection grating in optics; the grating
is replaced by the regular array of atoms and the light waves are replaced by
electron waves. Maximum response is obtained when the reflections add in
phase, that is when the condition n is an integral number, d is the
lattice spacing, and λ is the wave-
nλ = d sin θ (2.7)
length to be determined as a func-
is satisfied. tion of electron-gun accelerating
From eqn (2.7) the difference in angle between two successive maxima is voltage.
of the order of λ/d. Thus, if the wavelength of the radiation is too small, the
maxima lie too close to each other to be resolved. Hence, for good resolution,
the wavelength should be about equal to the lattice spacing, which is typically ∗ Planck (Nobel Prize, 1918) introduced
a fraction of a nanometre. The electron velocity corresponding to a wavelength this quantity in 1901 in a theory to ac-
of 0.1 nm is count for discrepancies encountered in
the classical picture of radiation from
h 6.6 × 10 –34 –1 –1 6 –1 hot bodies. He considered a radiator
v = = Jskg m =7.25 × 10 ms . (2.8) as an assembly of oscillators whose
mλ 9.1 × 10 –31 × 10 –10 energy could not change continuously,
but must always increase or decrease
The accelerating voltage may be obtained from the condition of energy
by a quantum of energy, hf.Thiswas
conservation the beginning of the twentieth century
for science and science has not been
2
1 mv = eV,
2 the same since. The confidence and as-
surance of nineteenth-century physicists
whence disappeared, probably forever. The most
6 2
mv 2 9.1 × 10 –31 (7.25 × 10 ) 2 –2 –1 we can hope nowadays is that our latest
V = = kg m s C = 150 V. (2.9) models and theories go one step further
2e 2 × 1.6 × 10 –19 in describing Nature.
The voltages used by Davisson and Germer were of this order.