LEAI-62 Electron Diffraction Apparatus
Lambda Scientific Systems, Inc
16300 SW 137th Avenue, Unit 132, Miami, FL 33177, USA
Phone: (305) 252-3838; Fax: (305) 517-3739
E-mail: sales@lambdasys.com; Web: www.lambdasys.com
COPYRIGHT
V1
COMPANY PROFILE
Lambda Scientific Systems, Inc. specializes in developing and marketing scientific
instruments and laboratory apparatus that are designed and manufactured specifically for
experimental education in physics at colleges and universities.
We supply high-quality, reliable, easy-to-use, and affordable scientific instruments and
laboratory apparatus to college educators and students for their teaching and learning of
both fundamental and advanced physics principles through hands-on experiments and
comprehensive instruction materials. Our products cover an extensive range of physics
experimental kits and experimental instruments, spectroscopic instruments, as well as
laboratory light sources and opto-mechanic components. Subjects include mechanics, heat
& thermodynamics, electromagnetics, optics, and advanced physics. All products come
with detailed teaching materials and experimental instructions/manuals.
Our spectroscopic instruments cover various general-purpose spectrometers/ spectrophotometers such as UV/Vis spectrophotometers, laser Raman spectrometer, dual-beam IR
spectrophotometer, FT-IR spectrometer, CCD spectrometer and monochromators. We also
provide specially designed spectroscopic systems for teaching or demonstrating the
principles of spectroscopic instruments, such as modular multifunctional spectrometer and
experimental Fourier transform spectrometer.
Our light sources include Xenon lamp, Mercury lamp, Sodium lamp, Bromine Tungsten
lamp and various lasers.
We also provide a variety of opto-mechanical components such as optical mounts, optical
breadboards and translation stages. Our products have been sold worldwide. Lambda
Scientific Systems, Inc is committed to providing high quality, cost effective products and
on-time delivery.
CONTENT
1. Introduction ................................................................................................................................ 1
2. Theory ........................................................................................................................................ 1
3. Apparatus Structure and Specifications ..................................................................................... 4
4. Safety Warning........................................................................................................................... 5
5. Operational Procedures .............................................................................................................. 5
6. Experimental Contents ............................................................................................................... 6
7. Examples of Data Recording and Processing ........................................................................... 8
1.
Introduction
Back in the early 20th century, it was known that light has wave-particle duality. In 1924, De
Broglie proposed a hypothesis that all micro particles have wave-particle duality. In 1927,
Davisson and Germer performed experiments of electron diffraction reflected from crystalline
nickel and verified the wave nature of electrons. In the meantime, Thomson independently
acquired a diffraction pattern of electrons passing through a crystal film, which further verified
the de Broglie hypothesis of wave-particle duality, and the wavelength of a de Broglie wave
was measured. At present, electron diffraction has become an advanced technology to study the
crystal structure of solid thin films and surface layers.
2. Theory
A schematic of the electron diffraction tube of the apparatus is shown in Figure 1. A circular
crystalline metal film target with a diameter of 15 mm is placed between the electron gun and
the screen in the diffraction tube. Electron diffraction pattern is formed by the direct hitting of
an electron beam on the crystal surface.
Figure 1 Schematic of electron diffraction tube
1
In the electron gun portion, electron beam is emitted from the cathode which is heated by the
filament, is further accelerated by an electric field of approximate 20 kV. After electrostatic
focusing and deflection, the electron beam is focused onto the target surface.
If an electron beam passing through the crystal film at speed v, the de Broglie wavelength of
the electron beam is:

h
h

p mv
(1)
where h is Planck’s constant, p  mv is the momentum of the electron, m is the mass of the
electron. Since the kinetic energy of the electron is:
1 2
mv  eV
2
(2)
where e is the charge of the electron and V is the accelerating voltage, the de Broglie
wavelength of the electron beam can be rewritten as:

h
2meV
(3)
Since m=9.109×10−31 kg, e=1.602×10−19 C, and h=6.626×10-34 m2 kg/s, we get:
 1.50 
 

 V 
1/ 2
(4)
where  is in unit of nanometer (nm), and V is in unit of volt (V). Since atoms are aligned
regularly in a crystal, they form a series of parallel planes in various directions with each serial
parallel plane presented by Miller indices (h, k, l). By considering an electron beam hitting on a
serial parallel plane of the atom structure, as shown in Figure 2, the condition of the beam
exiting from the film is:
n  2d sin 
(5)
where n is an integer,  is the angle between the diffracted direction and the incident direction
of the electron beam, and d is the distance between two adjacent parallel planes.
2
Figure 2 Schematic of electron beam diffraction from a crystal film
If  is small, sin can be approximated by =r/2D, where r is the radius of the diffraction ring,
and D is the distance between the target and the screen.
The distance between two parallel planes of Miller indices (h, k, l) is:
a
d
(6)
1
2 2
(h  k  l )
2
2
where a is the lattice constant, i.e. the edge length of a single crystal cell. By substituting (6)
into (5), we get
2a sin 

(7)
1
2 2
n( h  k  l )
2
2
If let H  nh , K  nk and L  nl , we have
2a sin 

1
2 2
(H  K  L )
2
2

r

D
a
1
2 2
(8)
(H  K  L )
2
2
This means that the nth order Bragg diffraction of any plane with Miller indices (h, k, l) can be
considered as the first order Bragg diffraction of plane (H, K, L).
Alternatively, one can abolish the small angle approximation by using following sine function
in Equation (8)
3
Electron wavelength can be acquired by using Equation (3) or Equation (8) from which the
crystal lattice constant or the Miller indices of a specific diffraction ring can be determined.
For face-centered cubic crystals such as gold and aluminum, geometric structure factors
determine that only planes of all even-numbered or odd-numbered Miller indices can generate
diffraction patterns, while other planes have zero reflection. Thus, Miller indices for generating
diffraction rings are limited as shown in the table below.
h
k
l
h2  k 2  l 2
1
2 2
(h  k  l )
2
2
111
3
1.732
200
4
2.000
220
8
2.828
311
11
3.316
222
12
3.464
400
16
4.000
331
19
4.358
3. Apparatus Structure and Specifications
This electron diffraction apparatus consists of a diffraction tube and a high voltage source.
The structure of the diffraction tube is shown in Figure 1. The high voltage applied between the
target (crystal thin film) and the cathode is 0~20 kV continuously adjustable, which is
displayed by a digital meter on the panel. The cathode, filament, and other anodes are powered
by other sources.
Main specifications are as follows:
High voltage:
DC 0-20 kV adjustable
Filament:
voltage 6.3 V, current 0.8 mA
Diffraction sample:
Gold (Au)
Distance between target and screen: shown on the side panel of the apparatus
Screen diameter:
130 mm
Overall dimensions:
360 mm  200 mm  500 mm
4
4. Safety Warning
1) Up to 20 kV high voltage (lethal) is used by the apparatus, do not open the cover or
touch internal metal pins or leads when high voltage is on. Make sure the apparatus is
well grounded by using the provided grounding wire (i.e. connect the “ground terminal”
on the back panel to a grounded site of the lab).
2) Turn high voltage to zero when not observing the diffraction pattern to minimize
possible exposure to the x-ray from the apparatus. Also, do not maintain the screen at a
high brightness state for a long period of time, otherwise the lifetime of the screen
would be reduced.
3) To minimize possible exposure to the x-ray from the apparatus, do not operate the
apparatus with a high voltage for an extended period of time.
4) Avoid using the apparatus near any strong magnetic fields.
5) Although the apparatus can provide a high voltage up to 20 kV, it is recommended to
complete experiments at lower voltage to extend the lifetime of the electron tube.
5. Operational Procedures
1) Prior to turning on power, turn the high voltage knob count-clockwise to minimum, to
prevent possible tube damage if high voltage is applied before warm-up.
2) Turn on power and preheat for 60 seconds.
3) Set high voltage to round 6.0 kV, slowly adjust brightness, focus and auxiliary focus,
until a light spot is observed on the screen. (If still no light spot is observed, try to
adjust X-position and Y-position knobs concurrently.) Note: the X- and Y-position
controls do not correspond to conventional X and Y axes, as these controls are not
independent of each other but are correlated, so they should be used in a combined
manner to steer the electron diffraction pattern to a proper location on the screen.
4) Adjust X-position and Y-position knobs to bring the pattern to a proper location on the
screen to achieve the best effect. Note, in order to extend the lifetime of the diffraction
tube, it is preferred to set brightness at a moderate level. Note: do not remain the bright
spot at a fixed location on the screen for a long time. When not observing pattern, turn
high voltage to zero to preserve the lifetime of the electron tube.
5) Before doing measurement, wait a few minutes to allow the system to be warmed and
stabilized. During experiment, increase high voltage slowly. After experiment, turn
5
high voltage knob to minimum location.
6) Turn off power.
Figure 3 Schematic of front panel
6. Experimental Contents
A. Measure electron wavelength and verify de Broglie relation
Use the provided ruler to measure the radii of the diffraction rings on the screen at different
accelerating voltages. Distance D from the target to the screen is given (on the back panel of
the apparatus). Accelerating voltage can be read from the meter. Starting from 12 kV or higher
(note: this start voltage may vary from batch to batch of the CRT), increase the accelerating
voltage at a step of 1 kV while measuring the ring radius r of the same order at each
accelerating voltage. Substitute r and D into Equations (3) and (8), respectively, calculate de
Broglie wavelength, and compare the two results (since the target in this apparatus is gold, its
lattice constant is a=0.40786 nm).
B. Measure crystal lattice constant
Measure the diffraction ring radii of planes (1 1 1), (2 0 0), (2 2 0), (3 1 1), (4 2 2), …,
respectively, at different accelerating voltages (14 kV, 16 kV, 18 kV, and 20 kV), as shown in
Figure 4. Use Equations (3) and (8) to calculate the lattice constant of gold.
6
Figure 4 Assignment of electron diffraction rings
C. Measure Miller indices
Based on experiments A and B, design your own experiment to measure the Miller indices of
the corresponding diffraction rings.
D. Calculate Planck’s constant
According to experiment A, draw graph of 2 ~
1
, and derive Planck’s constant.
V
Precautions:
1. Only plastic rulers should be used to measure the diffraction rings, do not use a steel ruler or
caliber for measurement; as otherwise the fluorescent screen of the CRT could be scratched.
2. When relocating the apparatus, handle it gently. The apparatus should be isolated from
mechanical shocks to avoid damage to the fragile glassware (CRT).
3. Under normal conditions, the CRT can last many years to come; however, it could crack or
even break under the impact of mechanical shock causing vacuum leak inside the glass tube.
To help diagnose if the vacuum inside a CRT is leaked, specific dye in black color was pre-
7
painted on specific areas inside each CRT. If the black dye turns white as shown in the photo
below, the CRT has lost vacuum and it is deemed as a broken one which needs a replacement.
Figure 5 Photo of leaked CRT
7. Examples of Data Recording and Processing
Note: the data below are for reference purposes only, not the criteria for apparatus performance
In Tables 1 and 2, equation (8) is used to calculate the wavelength of electrons from various
rings whereas D=256 mm, and a=0.40786 nm.
Table 1 Wavelength of electrons at an accelerating voltage of 12 kV
1
2 2
hkl
(h  k  l )
r (mm)
 (nm)
111
1.732
10
0.920
200
2
11.6
0.924
220
2.828
18.5
1.042
311 (222)
3.316
21.2
1.018
400
4
28
1.115
331 (420)
4.358
32.2
1.177
422
4.9
37
1.203
511 (333)
5.196
38.6
1.183
2
2
8
The averaged wavelength in Table 1 is 1.073 nm, as compared to the theoretical value of 1.118
nm derived from equation (4) yielding an error of roughly 4%.
Table 2 Wavelength of electrons at an accelerating voltage of 16 kV
1
hkl
(h2  k 2  l 2 ) 2
r (mm)
 (nm)
111
1.732
9.8
0.900
200
2
11.1
0.884
220
2.828
15.8
0.890
311 (222)
3.316
18.6
0.896
400
4
24.2
0.964
331 (420)
4.358
28.5
1.042
422
4.9
32.1
1.042
511 (333)
5.196
33.7
1.033
The averaged wavelength in Table 2 is 0.956 nm, as compared to the theoretical value of 0.968
nm derived from equation (4) yielding an error of roughly 1%.
Lambda Scientific Systems, Inc will take no liability for any injury or damage that may be caused by improper
handling and/or incorrect using of the electron diffraction apparatus. Lambda Scientific Systems, Inc assumes
no responsibility for the accuracy and/or the suitability of the information provided in this manual.
9