ELECTRON DIFFRACTION

ELECTRON DIFFRACTION

Electrons : wave or quanta? Measurement of wavelength and momentum

of electrons.

Introduction

Electrons display both wave and particle properties. What is the relationship between the elec-trons’ wavelength and their momentum?

Theory

Louis de Broglie developed the idea that electrons exhibit both particle and wave properties. He proposed that, analogous to photons, the wavelength λ of the electron wave is given by

λ = h p =

h

mv (1)

where h is Planck’s constant and p is the electron momentum which is given by p = mv in the non-relativistic limit.

In order to observe the wave nature of electrons one directs a beam of electrons through a thin layer of carbon and analyzes the angles at which electrons emerge. If electrons behave like particles, the distribution of electrons varies continuously as a function of angle as indicated in Fig. 1. This distribution will vary only slightly with changes in electron energy.

counts electron beam

carbon target

Theta

Figure 1: Electrons as Particles

Continuous distribution of electrons as a function of angle.

The picture is different if electrons behave like waves. Analogue to the diffraction of x-rays (photons with wavelengths of a few ˚A) by crystals, the crystal may be considered to be a 3 dimensional diffraction grating for electron waves. The diffraction maxima are deduced apply-ing the concept of reflectapply-ing planes in the crystal. The locations of atoms in the two-dimensional

Hexagonal d d Incoming Beam d Cubic d d

Figure 2: Atomic Planes of Cubic and Hexagonal Crystals.

Two-dimensional slices of cubic and hexagonal crystals.

Different angles of reflection are indicated in the cubic crystal structure.

slices of cubic and hexagonal crystals are shown in Fig. 2. The entire crystal can be visualized as a stack of slices, the various lines representing planes intersecting the actual crystal.

Θ d sin( ) d Θ Θ Θ Θ

Figure 3: Electron Waves Reflected from Atomic Planes.

Atomic planes are treated as reflecting surfaces and allow waves from successive planes to interfere with one another. If the difference in path length for waves reflected from successive planes is an integral number of wavelengths (nλ, cf.F ig.3), constructive interference of matter waves occurs at angles for which

nλ = 2d sin Θ (2)

holds.

specific geometry of a crystal. In order to measure diffraction angles as a function of wavelength for a single crystal, one needs to vary not only the wavelength (λ) but also the orientation to find the diffraction maxima. While changing the electron wavelength is easy to obtain by changing the acceleration potential (v =q2eVa/m and Eq.1), varying the orientation of the crystal inside

an evacuated tube is difficult. In addition single crystals are difficult to obtain.

An easily realizable modification for a diffraction experiment is shining an electron beam of

Theta

electron beam

carbon target

counts

Figure 4: Constructive interference of an electron beam though a polycrystal.

given energy into a polycrystal. A polycrystal is a collection of micro-crystals (very small crys-tals) where each crystal is randomly oriented.

Most of the micro-crystals will not satisfy Eq. 2 for any of their atomic planes, but some are ori-ented in a way to allow constructive interference. Since the whole setup is symmetric according to the beam axis, the diffracted electrons will produce a conical shell of constructive interference for each atomic plane and each order (n). The most important atomic planes thereby will dom-inate the diffraction picture, which (using the apparatus described below) will show concentric rings, one for each atomic plane and order.

In effect, using a polycrystal makes it unnecessary to vary the orientation, as all plane orientations already exist among the micro-crystals. A typical pattern produced is shown in fig. 4.

Apparatus

An electron diffraction tube consisting of an electron gun, a carbon target, and a luminescent screen is used for this experiment (see Fig. 5). With this special vacuum tube electrons are produced and accelerated, and their diffraction on a carbon layer (polycrystal) can be studied.

Electron Gun

The heated cathode and anode make up the electron gun. Electrons are produced by heating a fil-ament that is located inside an oxide-coated metal ”can” called the cathode, electrons are ejected by thermionic emission from this heated piece of metal (see fig. 5).

Once emitted, the electrons are accelerated by two pairs of anode rings with adjustable acceler-ation potential Va(2000− 5000V ) (kilovolt power supply between the cathode (−) and anode

(+)) and form an electron beam.

Each electron in the beam has a kinetic energy equal to the accelerating electric potential (eVa)

times the electron charge (e). In the non-relativistic limit (eVa<< mc2), the electrons acquire a

kinetic energy of 1 2mv 2 _{= eV} a . (3)

Carbon Target

As the electron beam passes through the anode, it meets a very thin mesh containing vaporized graphite (carbon). The carbon suspension acts as the polycrystal described above. Diffraction caused by the mesh can be neglected.

Luminescent screen

Leaving the target, diffracted electrons travel a certain distance (L) and strike a phosphor screen. The beam now appears as concentric rings around a bright center. This pattern can be visualized as a set of one-dimensional diffraction patterns of bright spots rotated about the axis of the electron beam (see fig. 4).

As the screen is located inside a sphere, it is necessary to make geometric corrections to data. As value for the distance L between the carbon target and the screen take L = 148.5_{± 0.5mm and} the radius of the tube R is R = 65.0_{± 0.2mm. A sketch of the tube is shown in fig.6.}

Procedure

Equipment Needed

Equipment Notes

• A darkened room will help you see the diffraction patterns.

• Use high voltage cable between the anode (side plug on diffraction tube) and the electro-meter.

• Make sure power is turned off before handling connections. A --

+

+

Figure 5: Electron Diffraction Experimental Diagram.

Connections

1. Connect the electron diffraction tube to the power supply as shown in fig. 5.

2. Connect the electro-meter with the cathode and the negative port of the power supply.

Power Up

• Set the electro-meter at the sensitivity of 600µA.

• Make sure that the high voltage supply is set to minimum. Switch on the power and wait one minute for the cathode to reach a stable temperature.

Controlling the Current

The normal operation current should be kept below 150µA during measurements at voltages below 4.0kV . When it is necessary to go above this range, measurements should be made quickly and high voltage should be decreased as soon as results are obtained.

—————————————————————————————–

Caution: DO NOT EXCEED 200µA of current between cathode and anode! If you do, turn off high voltage immediately!

Watch the carbon target! If it starts to glow turn of high voltage immediately! —————————————————————————————–

Slowly increase the high voltage until you observe a pattern (_{≈ 2kV ). Monitor the current and} take precautions when exceeding the normal (150µA) range.

Measurements

Take readings of accelerating voltage vs. ring diameter for each observed ring. Measure the diameter arc length (diameter bent around the screen) using a flexible, transparent ruler (provided by lab).

Accelerating Voltage vs. Ring Diameter

• Start at minimum voltage and record diameters and voltages at 0.1kV increments for each ring.

At voltages above 4kV , take each reading quickly and decrease the voltage to zero after the measurements.

Geometric Analysis

• The distance L from the carbon target to the screen is 148.5 ± 0.5mm. The screen has a radius of curvature of 65mm. As this is not equal to L, one needs to correct the diameter measured. One may also take the index of refraction of the glass (n = 1.5) into account, but this second correction is not really worth the effort.

Analysis

1. Substitute Eq. 3 into the de Broglie relation Eq. 1 and solve for the de Broglie wavelength λ. Show that the resut gives m (meter) as unit.

2. Use the diffraction Eq. 2, the result obtained in the previous question and geometry (see fig.6)to find a relation between ring diameter D = 2r, target-screen separation L, and accelerating voltage Va(assume L = l).

3. Using geometry, find the ring diameter D in terms of the arc length. Drop the assumption L = l used above and find an expression for l = l(arclength). Write the previosly derived formula (No.2) incorporating the expressions for D and l.

L carbon target F sc_re e n a curvature R=65 mm r r l R l´ m e_a su re_d d ia m e te r 1.5 mm

Figure 6: Geometric Layout of the Electron Diffraction Tube

Method A:

• Show that if you assume sin Φ = Φ and l = L, the ring diameter D = 2r is proportional to 1/√Va. Find the constant of proportionality.

• Make a plot 1/√Vaversus D = 2r for each ring. Does a straight line fit your data? If so,

use the values of the slope to calculate the plane separation d that corresponds to each ring. Discuss the errors of your raw data. Sketch them in your graph. Estimate the error due to the approximations.

Method B:

• For each ring, calculate the plane separation d for each Va. Use a equation similar to

method A, but do not assume l = L. Find the mean and standard deviation.

Questions

1. What is the total direction change Φ in terms of Θ? Which order of diffraction (i.e., value of n) should be visible at smallest Φ.

2. If the crystal structure of carbon is cubic, what would be the spacing of the atoms? (Assume a density of 2.0g/cm3 _{and note that 12 grams of carbon contain Avogadro’s number of}

atoms). What acceleration potential Va(approximately) should be applied to the electrons

3. In principle an analysis of the ring diameters can tell what the crystal structure is like hexagonal or cubic, and what the atomic spacing is in the carbon. Fig.2 shows the com-plexity of this analysis. One must be satisfied with obtaining the correct order of magnitude for the spacing and showing that the electrons have a wavelength λ = constant/p; where the constant is something close to h, Planck’s constant. Explain how the data shows this to be true. If the energy of a photon is E = hf , why is its momentum p = h/λ?

4. If the kinetic energy of a particle exceeds 0.1mc2, its behavior must be described by rela-tivistic equations. What acceleration potential Va would be needed to produce relativistic

electrons? How does this potential compare to the maximum potential (_{≈ 5kV )?}

5. In a diffraction experiment with ordinary light (wavelengths of order 5000 ˚A or 5_×10−7m), a grating is made by scratching thin lines on glass. Why doesn’t this work well for electrons or x-rays? Why is a typical crystal the best choice (see Question 2)?

6. Consider the mesh in the experiment. Suggest several methods that the tube designers could have used to ensure that diffraction in the mesh is negligible (i.e. electrons it diffracts produce a faint or invisible pattern).