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Chemistry 120A, Spring 2009
Problem Set 1 Solutions
Due January 30, 2009
Problems
1. The DavisonGermer experiment of the diﬀraction of electrons by an ordered metal surface
gives a beautiful illustration of the validity of the de Broglie relation connecting the particle
like properties of electrons (their momentum) to their wavelike properties (their wavelength,
which causes diﬀraction if same as interatomic spacing)
(a) A 40 eV beam of electrons shows no diﬀraction from a silver surface, a 54 eV beam
exhibits a clear diﬀraction patter. By 60 eV the diﬀraction structure is again largely
lost. Work out the wavelengths that these three beam energies correspond to.
We know from de Broglie that
λ
=
h
p
,
and
KE
=
1
2
mv
2
=
p
2
2
m
Therefore,
p
=
√
2
mE
and
λ
=
h
√
2
mE
For 40 eV
λ
=
6
.
626
×
10

34
Js
r
2
·
9
.
109
×
10

31
kg
·
40 eV
·
1
.
602
×
10

19
J
1
eV
λ
= 1
.
9393
×
10

10
m =
.
19393 nm
For 54 eV
λ
=
6
.
626
×
10

34
Js
r
2
·
9
.
109
×
10

31
kg
·
54 eV
·
1
.
602
×
10

19
J
1
eV
λ
= 1
.
6691
×
10

10
m =
.
16691 nm
For 60 eV
λ
=
6
.
626
×
10

34
Js
r
2
·
9
.
109
×
10

31
kg
·
60 eV
·
1
.
602
×
10

19
J
1
eV
λ
= 1
.
5834
×
10

10
m =
.
15834 nm
1
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View Full Document(b) If one were worried about damage to the surface caused by the energy of the electrons
necessary to cause diﬀraction, one could consider using a beam comprised of diﬀerent
particles. Work out the energy of a beam of helium atoms that could exhibit diﬀraction
on the same surface and discuss whether or not this is a gentler probe.
In order to get diﬀraction, we need a wavelength of 1.6691
˚
A, as that is the wavelength
of the 54 eV electron beam.
Using
λ
=
h
p
and
E
=
p
2
2
m
we get
E
=
1
2
m
·
±
h
λ
²
2
E
=
1
2
m
He
·
³
h
λ
´
2
=
1
2
·
6
.
6423
×
10

27
kg
·
6
.
626
×
10

34
Js
1
.
6691
×
10

10
m
!
2
= 1
.
1864
×
10

21
J
·
1 eV
1
.
602
×
10

19
J
=
.
007406 eV
The beam of Helium atoms has a lower energy, and therefore will be a gentler probe of
the surface.
2. Consider an electron that is initially far away from a proton and has no kinetic energy.
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 Spring '09
 HEADGORDON

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