Student Name and I.D.: Peter
Lab session: Lab 2B
Submission
Date: 4-4-2001
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1.
Introduction
In
1926, at the Bell Telephone Laboratories, Clinton Davison and Lester Germer were
investigating the reflection of beams from the surface of nickel crystals. A
beam of electrons from a heated filament is accelerated through a potential
difference V. Passing through a small aperture, the beam strikes a single
crystal of nickel. Electrons are scattered in all directions by the atoms of the
crystal and some of them striking a detector.
If
we assume that each of the atoms of the crystal can act as a scatter, then the
scattered electron waves can interfere, and we have a crystal diffraction
grating for the electrons.
The purpose of this experiment is to investigate the wave nature of the electron through the diffraction of electron beams. In this experiment, a graphite sample consists of atoms in the hexagonal arrangement shown in Figure 1.
Figure 1: Hexagonal arrangement of atoms
in a graphite sample
2.
Theory
In quantum theory, de Broglie proposed that particles may be equivalently described as waves. The wave-particle duality is summarized by the relations between energy (E), frequency (v), momentum (p) and wavelength (£f): E = hv and p = h / £f.
The
diffraction of electrons was experimental evidence of the wave nature of matter.
The kinetic energy of an electron which is accelerated across a potential
difference V is given by
mv2 = eV. Its wavelength is given by ¡K
Diffraction
by a crystal:
Figure 2: Diffraction by a crystal
Diffraction by a crystal is due to the interference of waves reflected from parallel planes of atoms as shown in Figure 2. Then
2 d sin £c = n £f n = 1, 2, 3, ¡K --------------------- [2]
where
d is the interlayer spacing, and £c is the reflecting angle.
Powder pattern:
Figure 3: Powder pattern
In Figure 3, the diffraction rings are concentric about the central undiffracted beam. The inner ring arises from plane separated by d10, while the outer ring comes from separated by d11. The transmission angle £c¡¦ is given by
sin £c¡¦ = D / (2L)
sin 2 £c = D / (2L) ----------------------- [3]
where
£c = diffraction angle and L = 0.135m.
Consider Equation [2], putting n = 1, we have
£f ~ 2 d sin £c ~ d sin £c¡¦ ----------------------- [4]
Combining Equation [1] and [4], we have
V ¡V1/2 = Dd / (24.6L)
---------------------- [5]
3.
Procedure
Figure 4: Experimental Setup
The
electrical connection was made as shown in Figure 4. The 6.3V output from the
Kilovolt power supply was used for VF. The external bias connections
was made to the low voltage while the accelerating voltage (VA) was
connected to the Kilovolt supply. The Kilovolt Power Supply was then switched on
and the output was adjusted to zero. The filament began to glow and was waited
for one minute before adjusting the accelerating voltage.
The
bias voltage was set to about 20V and the accelerating voltage was set to 3.0kV.
The bias voltage was slowly decreased until the diffraction rings were sharp to
be seen, and the emission current was kept below 100£gA. The diameter of the
diffraction rings was measured as a function of accelerating voltage in steps of
500 volts from 2.5kV to 5.0kV.
4.
Result and Analysis
Error in D = + 0.00025m
Error
in VA = + 0.05kV
The diffraction angle £c is calculated by equation [3].
Two
graphs are plotted. Figure 5 shows the graph of VA-1/2
against D (inner), and Figure 6 shows the graph of VA-1/2
against D (outer).
Figure 5: VA-1/2
against D (inner)
Figure 6: VA-1/2
against D (outer)
The
two graphs are both straight lines almost passing through the origin. The
interlayer spacings d10 and d11 can be determined from the
slopes.
d10 = slope x (24.6 L) x 10-10 = 0.5339 x 24.6 x 0.135 x 10-10 = 1.773 x 10-10 m
d11
= slope x (24.6 L) x 10-10 = 0.3591 x 24.6 x 0.135 x 10-10
= 1.193 x 10-10 m
The
experimental value of the wavelength for an electron is then calculated by
equation [4].
5.
Error Analysis
£G VA = + 0.5kV
£G
D = + 0.0005m
£G d10 = £G slope = 0.000225 x 10-11 m
£G
d11 = £G slope = 0.0000541 x 10-11 m
By Taylor series, sin £c = £c ¡V £c3 / 3!
Take
k=2, £G £c = £c3 / 3!
6.
Discussion
1)
A lattice with wider interlayer spacing can be used to remove the error
from small angle approximation.
2)
From the results, the experimental value of the wavelength of electrons
is close to the de Broglie wavelength. We can find the phenomenon of the
diffraction pattern from the observation of electrons. Thus we can say that
electrons have a wave-like property and consistent with the de Broglie theory.
3)
We assume that each of the atoms of the crystal can act as a scatter,
then the scattered electron waves can interfere, and we have a crystal
diffraction grating for the electrons. Because the electrons were of low energy,
they did not penetrate very far into the crystal, and it is sufficient to
consider the diffraction to take place in the plane of atoms on the surface. The
situation is entirely similar to using a reflection grating for light; the
spacing d between the rows of atoms on the crystal is analogous to the spacing
between the slits in the optical grating.
7.
Error Source
1) Error in the measured values of diameter is dominant error. Involving eye judgment cause significant error in measurement of diameter.
2) Error from small angle approximation
3)
The diffraction rings are not bright and sharp enough.
8.
Suggestions and improvements
Precautions:
1) Make sure the electrical connections of the circuit is correct before turning on power supplies
2) The emission current must be kept under 100£gA in order to protect the graphite sample. The sample should not glow red hot.
3) Do not leave a bright beam on the fluorescent screen when not doing measurement. Otherwise the screen can be burnt out.
4)
After finishing the experiment, adjust the high voltage to zero, and then
turn off the power supplies.
Improvement:
1) A lattice with wider interlayer spacing can be used to remove the error from small angle approximation.
2) Carefully measure the diameter of the rings.
3)
Repeat measurement for more times.
9.
Conclusion
From the observation of the diffraction of electrons, we can conclude that electrons have a wave-like property and consistent with the de Broglie theory. The interlayer spacing d10 and d11 are found to be respectively 1.773 x 10-10 m and 1.193 x 10-10 m.