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6 b. E LECTRON D IFFRACTION

(Adapted with permission from UC San Diego lab manual; updated by Scott Shelley & Suzanne Amador Kane

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GOALS

Physics

This experiment demonstrates that accelerated electrons have an effective wavelength, ,by diffracting them from parallel planes of atoms in a carbon film.

This allows you to measure the spacings between two sets of parallel planes of atoms ingraphite, a crystalline form of carbon.

The technique of electron diffraction is often used in current scientific research to studythe atomic-scale properties of matter, especially on surfaces and in biological specimens.

Techniques

Control the wavelength of the electron beam by varying the accelerating voltage. Use the De Broglie expression for the wavelength of the electrons and the Bragg

condition for analyzing the diffraction pattern.

Calculating the uncertainties in the data points on your graph gives you a goodopportunity to use the principles of error propagation.

References

Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei andParticles , Chapter 3-1 (sections on de Broglie waves and electron diffraction. On reserveand in the Physics Lounge, H107.)

BACKGROUND

Se%eral of your laboratory e periments show that light can e hibit the properties of eitherwa%es or particles' he wa%e nature is e%ident in the diffraction of light by a ruled grating and inthe interferometer e periments' )n these e periments* wa%elength* phase angle* and coherencelength of wa%e trains were in%estigated++all features of wa%e phenomena' ,owe%er* the

photoelectric effect cannot be e plained by a wa%e picture of radiation' )t re-uires a model inwhich light consists of discrete bundles or -uanta of energy called photons' hese photons

beha%e li.e particles' here are other e amples illustrating this dual nature of light' /enerally*those e periments in%ol%ing propagation of radiation* e'g' interference or diffraction* are bestdescribed by wa%es' hose phenomena concerned with the interaction of radiation with matter*such as absorption or scattering* are more readily e plained by a particle model' Someconnection between these models can be deri%ed by using the relationship between energy andmomentum for photons found from 0a well1s e-uations and special relati%ity2

6b-1

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Electron Diffraction 6b-2

pc E = ' 3-'4

)n this e-uation* 3 is the energy of a photon* c the speed of light in %acuum and p the photon1smomentum' 5rom the photoelectric e periment we learned that light may be considered toconsist of particles called photons whose energy is

hf E = 3-'"

where f is the fre-uency of light and h is 6lanc.7s constant' 8e may e-uate these two energiesand obtain2

hf pc = * or 3-'9a

hc

hf p == 3-'9b

where : is the wa%elength of the light' hus the momentum of radiation may be e pressed interms of the wa%e characteristic :'

his dual wa%e+particle model of radiation led de roglie in 4

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of electrons from a grating' ,owe%er* the spacings between the rulings in man+made gratings areof the order of se%eral hundred nm' 5rom e-uation (4$* we find that e%en with an accelerating%oltage as low as 4## ?* the electron wa%elength is only #'4" nm' As we will see shortly* such alarge difference between the grating spacing and the electron wa%elength would result in animmeasurably small diffraction angle' )t was recognized* howe%er* that the spacings between

atoms in a crystal were of the order of a few tenths of a nanometer' hus* it might be feasible touse the parallel rows of atoms in a crystal as the Bdiffraction gratingB for an electron beam' his possibility seemed particularly promising since it had been found that +rays could be diffracted by crystals* and +ray wa%elengths are of the order of the wa%elengths of 4## e? electrons'

)t was also .nown that atoms are regularly arranged in a crystal into a repeating spatial patterncalled a lattice' 5igure 4 shows some of the possible arrangements of atoms in a cubic lattice' (a$is the simple cubic form' 8hen an atom is placed in the center of the simple cube* we get (b$* the

body+centered+cubic form'

Figure 1 2 hree cubic arrangements of atoms in a crystal' (a$ simple cubic* (b$ body centered+cubic* (c$ face+centered cubic

When atoms are placed on the faces of the cube, as in Fig. 1 c), the arrangement is called face-centered-cubic. For example, the atoms in nickel and sodium chloride are arranged in the face-centered-cubic pattern. In an iron crystal, the body-centered-cubic arrangement is found. Figure2 shows a view of the atoms looking perpendicular to one of the cubic faces. Three differentorientations of parallel rows of atoms are distinguished with different spacings between theparallel rows. These parallel rows of atoms lie in parallel atomic planes and it is evident thatthere are a large number of families of parallel planes of atoms in a crystal. We will now showthat waves scattered from these regularly spaced planes of atoms within crystals can act togenerate constructive and destructive interference patterns similar to those generated by slits indiffraction gratings.

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Figure 2 2 )nterplanar spacings* d* of different families of parallel planes in a cubic array ofatoms'

Figure 3 2 Scattering of wa%es from a plane of atoms' 6ath difference for wa%es from ad acentatoms'

We consider the scattering of waves from a single plane of atoms as shown in Fig. 3. The atomsare spaced a distance d' apart. The incident wave makes an angle with a row of atoms in thesurface plane waves of atoms; ca is the wavefront. The scattered wave makes an angle withthe atom row; its wavefront is eb . Constructive interference will occur for the rays scatteredfrom neighboring atoms if they are in phase; if the difference in path length is a whole number ofwavelengths. The difference in path length is bcea . Therefore

md d bcea == coscos , where m is an integer. Another condition is that rays scatteredfrom successive planes also meet in phase for constructive interference. Figure 4 shows theconstruction for determining this condition.

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Figure 4 2 6ath difference for wa%es scattered from successi%e planes of atoms'

he difference in path length for rays tra%eling from planes 4 and " is seen to be cbba + * thee tra distance tra%eled by the ray scattered from plane "' his path difference must again be anintegral number of wa%elengths' herefore

nd d cbba =+=+ sinsin ' 3-'

hese conditions can be satisfied simultaneously if = ' )n that case m = # for the firstcondition and

sin" d n = for the second condition' 3-' E

his relation was de%eloped by ragg in 4

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Figure 2 he e perimental arrangement used by homson for his transmission electrondiffraction research'

5igure @(a$ shows a beam of electrons of wa%elength tra%eling from the left and stri.ing a plane of atoms in a crystallite' )f this plane ma.es the angle with the incident beam such that

sin" d = * where d is the spacing of successi%e atomic planes* the beam will be diffracted intothe angle with respect to the atom plane (or the angle " that the diffracted beam ma.es withthe incident beam$'

Figure 6 2 Showing how the randomly oriented crystallites in a polycrystalline film scatter into a

cone when the ragg condition is fulfilled by planes of atoms disposed symmetrically about theincident beam'

Fow there are many randomly oriented crystallites in this film' hus we may e pect thatthere will be crystallites in which this diffracting plane ma.es the same angle with the beamdirection but rotated around the beam in a cone as shown in 5ig'

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and transmission diffraction e periments' 5or this wor. /ermer and homson were awarded the Fobel 6rize in 4

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lattice position, or you may get a position that does not correspond to the actual honeycomblattice, but instead the empty center of the hexagons. To think about the honeycomb graphitelattice properly, you need to consider a unit cell that consists of TWO carbon atoms at a time, asshown in Fig. 8 by the two atoms connected by a solid line. The lattice vectors are shown on therighthand image.

Prelab Question 1: Satisfy yourself that if you take the two-atom unit cell indicated, you cangenerate the entire lattice by moving it along integer multiples of the lattice vectors.

Figure #" ,oneycomb lattice found within the layers of graphite' All nearest+neighbor carbon atoms (blac. circles$within the plane are connected by e-ui%alent chemical bonding* with a charcter intermediate between single anddouble bonds' Gefthand image2 the lattice* showing the unit cells (two atoms connected by a solid line$' Highthandimage2 ,oneycomb lattice* showing the unit %ectors needed to generate the lattice* using the unit cell indicated atleft' ' F' Ashcroft and D' 0ermin , Solid State Physics ' roo.s Cole* 4< @'

All this is relevant to your electron diffraction experiment (or any diffraction experimentwith x-rays, neutrons, etc.) because the lattice vectors and unit cells determine which atomicplanes are involved in Bragg diffraction. Only atomic planes separating adjacent unit cells willgenerate Bragg diffraction, because only those planes repeat exactly throughout the lattice. Thisis shown in Fig. 9(a) for graphite. The relevant d spacings are 0.123 nm and 0.213 nm. (Fig. 9(b)shows the distance between the stacked graphene planes. These are arranged so as to give Braggdiffraction with a distance d = 0.688 nm.) An easy way to see which planes will give Braggdiffraction is to replace the (confusing) honeycomb lattice with the simpler underlying latticecomposed of the locations of the pairs of atoms. Any planes drawn through this lattice will resultin Bragg diffraction.

Prelab Question 2: Draw the lattice formed by replacing the two-atom unit cell in Fig. 8 by asingle circle, and prove to yourself that the planes involving this new, simpler lattice have thespacings shown in Fig. 9.

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(a) (b)

Figure $ 2 (a$ Atom arrangements in graphite showing the two sets of planes within the graphenelayers that produce the diffraction rings you obser%e in your e periment' hese spacings are#'4"9 nm and #'"49 nm' (b$ he graphene planes are stac.ed as shown to form the 9D lattice'

Fote that the two layers ad acent to each other ha%e ine-ui%alent atomic arrangements' hismeans that the effecti%e lattice spacing for diffraction between layers is @EE pm (picometers$* asshown' http2!!phya'yonsei'ac'.r!Iphylab!board!e pJref!upfile!phywe! J4J49'pdf

Experimental roce!ure

Equi%&en'

1. Electron diffraction tube with graphite thin film target.2. Power supply that provides the current to heat the anode and the high voltage for

accelerating the electrons.

3. Calipers for measuring diffraction ring diameters.

CA(TION

The 5kV power source can give you a very nasty shock. Verify that your circuit iscorrectly wired before turning on power. Have your instructor check the circuit.

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Figure 1) 2 he electron diffraction tube

The electron diffraction tube is sketched in Fig. 10. The graphite film is mounted in the anode asshown. The variable accelerating voltage is provided by the 5kV dc supply. The electrons areemitted from an indirectly heated oxide coated cathode. They boil off this cathode filament wireswith a small thermal energy which is negligible compared to the kilo eV provided by theaccelerating voltage. The heater voltage is supplied by the same power supply as provides theaccelerating voltage. This power supply also supplies a negative voltage to the metal cansurrounding the cathode that emits the electrons. This serves to focus the electron beam. Thediffraction rings are viewed on the phosphor screen on the glass bulb. he apparatus should beconnected up' )f not ha%e your instructors do so' he connections are2 59 & 5> = filament%oltage (orientation not important$; A4 not connected; / Anode (red ,igh ?oltage connector$;

C Cathode (blac. ,igh ?oltage connector$'

After ha%ing your circuit chec.ed* start the e periment by allowing the heater current to stabilizefor about a minute before turning up the accelerating %oltage* on the front panel of the powersupply' ou can also read off the high %oltage* ?* from the front panel display' ou will seerings on the phosphor screen for selected %alues of the accelerating %oltage* but you will need todar.en the room before doing so' As discussed abo%e* 5ig' < shows the arrangement of theatoms in a graphite crystal' hey are located on the corners of a he agon and two principalspacings of the atom planes are indicated as d 4 and d " ' As you turn up the accelerating %oltage*you will see two rings on the screen* as shown in 5ig' 44 below' 3ach ring corresponds to one ofthe graphite d spacings from 5ig'

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should not be satisfied by ust a few either' 0a.e a rational choice of how many differentaccelerating %oltages to use for each distinct ragg diffraction ring* and e plain your reasoning'

nce you are done* record your data for the two different layer spacings (ring 4 and ring "$ as afunction of ring diameter* D* %s' %oltage* ?' his is your basic dataset you will now analyze todetermine the graphite layer spacings'

Fote that in our earlier discussion and in 5ig' 44* we define the angle between the direction ofthe undiffracted electron beam and the diffracted ring as " ' his con%ention is common inthe diffraction and crystallography literature* but it can be confusing' e sure to note thisfactor of two in case you wind up with a missing factor of two in your deri%ationsN

Figure 11 2 S.etch of the geometry in%ol%ed in determining scattering angle* "-* from the measured ring diameter*

D' (Fote that the @@'# mm is supposed to be the radius of the spherical glass electron diffraction bulb' D is the

diameter of the ring you measure on the glass electron diffraction apparatus* while D1 is the diameter of the larger

ring you would get by e trapolating the path of the electron1s past the bulb onto a flat screen tangential to the front

of the bulb'$

As e plained abo%e* the ragg diffraction condition for the polycrystalline graphite film is

sin" d = * 3-'

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