Page 264 - Essentials of physical chemistry

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226 Essentials of Physical Chemistry

s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
r ﬃﬃﬃﬃﬃﬃ
mV 2 2E 2(5:0 eV)(1:60217653 10 19 J=eV) 6
¼ 1:36205154 10 m=s
E ¼ ) V ) ¼ 31
2 m 9:1093826 10 kg
h 6:62609 10 34 J s 10
¼ 5:4847606 10 m or about
mV (9:1093826 10 kg)(1:36205154 10 m=s)
l ¼ ¼ 31 6
l ﬃ 5.485 10 8cm ¼ 5.485 Å, which is comparable in size to a molecule. Thus, wave mechanics
is useless for macroscopic calculations but essential for molecular calculations!

DAVISSON–GERMER EXPERIMENT

Following the 1923 paper by De Broglie [5], a number of experimental specialists tried to verify the
wave properties of particles. A group at Bell Laboratories designed an experiment to test the
De Broglie idea using low energy electrons. Several papers were published but the most detailed
appeared in 1927 by Davisson and Germer [10]. Clinton Davisson (1881–1958) was awarded the
Nobel Prize for this work in 1937. The initial experiments consisted of scattering a beam of
electrons with a uniform low energy at various angles off of small blocks of platinum and nickel.
These experiments gave just broad angular patterns and did not follow De Broglie’s equation.
In 1925, there was an accident at the Bell Laboratories site in which a liquid air container
exploded and broke the glass enclosure of the experiment while a small block of nickel was being
bombarded by an electron beam in a vacuum. When air rushed into the experimental chamber, a
layer of oxide formed on the surface of the nickel block, which was very hot from the electron
impacts. That was a serendipitous accident because after repairs were made, the block had to be
heated for a long time under vacuum to vaporize the oxide layer. An electric heater under the block
brought the temperature of the block close to the melting point of the nickel under vacuum for
perhaps a month or more. At the end of that time the polycrystalline nickel had large areas of
annealed crystalline nickel clearly visible. When the electron beam experiment was restarted, the
angular pattern of the scattered electrons showed sharp peaks and eventually some 30 peaks were
assigned to scattering off various planes of the face-centered cubic (fcc) nickel crystal surface. If you
draw an fcc structure as in Figure 10.10 you can see that you could slice through the lattice in a
number of ways, and the {1, 1, 1} plane is formed through the corners (a, a, a) opposite to (0, 0, 0)
where a is the unit cell dimension. Many planes can be deﬁned in terms of the unit cell dimension
such as {1, 1, 1}, {1, 1, 0}, and {1, 0, 0} where the indices refer to the (x, y, z) dimensions. In their
paper, Davisson and Germer say they oriented the surface of the Ni crystals perpendicular to the
{1, 1, 1} planes as in Figure 10.12 and apparently used a spacing between the planes of 2.18 Å.
A modern value of the fcc structure of nickel shows the a parameter to be 3.5238 Å [11], and the
formula for the spacing between {1, 1, 1} planes is

a 3:5238 A ˚
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2:03447 A ˚
d lmn ¼ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ p
2
2
l þ m þ n 2 1 þ 1 þ 1
The modern value of the Ni (fcc) unit cell makes the angle of the diffracted wave slightly higher
than their 508 conclusion but there are uncertainties associated with the angle of the surface to the
electron beam and whether the whole surface was pure {1, 1, 1} (Figure 10.13). Still the calculated
angle agrees with the overall pattern of the scattering peaks shown in their paper. Most of the
scattering peaks corresponded to the well-known x-ray scattering patterns that strengthen the wave
analogy! Note they speciﬁcally state that they used nl ¼ 1d sin(u), which makes sense according to
the diagram shown in Figure 10.14 for a perpendicular beam compared to the grazing incidence
equation of nl ¼ 2d sin(u).

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