Chapter 36Scattering of Waves

CHAPTER 36 SCATTERING OFWAVESWe will briefly interrupt our discussion of the hydrogenatom and study the scattering of waves by atoms. It wasthe scattering of electron waves from the surface of anickel crystal that provided the first experimentalevidence of the wave nature of electrons. Earlierexperiments involving the scattering of x rays hadbegun to yield detailed information about the atomicstructure of crystals.

Our main focus in this chapter will be an experimentdeveloped in the early 1960s by Harry Meiners at R .P.I., that makes it easy for students to study electronwaves and work with de Broglie's formula λ = h/p .The apparatus involves the scattering of electrons froma graphite crystal. The analysis of the resulting diffrac-tion pattern requires nothing more than a combinationof the de Broglie formula with the diffraction gratingformula discussed in Chapter 33. We will use Meiner'sexperiment as our main demonstration of the wavenature of the electron.

36-2 Scattering of Waves

SCATTERING OF A WAVEBY A SMALL OBJECTThe first step in studying the scattering of waves byatoms is to see what happens when a wave strikes asmall object, an object smaller in size than the wave-length of the wave. The result can be seen in the rippletank photographs shown in Figure (1). In (1a), anincident wave is passing over a small object. You cansee scattered waves emerging from the object. In (1b),the incident wave has passed, and you can see that thescattered waves are a series of circular waves, the samepattern you get when you drop a stone into a quiet poolof water.

If the scattering object is smaller in size than thewavelength of the wave, as in Figure (1), the scatteredwaves contain essentially no information about theshape of the object. For this reason, you cannot studythe structure of something that is much smaller than thewavelength of the wave you are using for the study.Optical microscopes, for example, cannot be used tostudy viruses, because most viruses are smaller than thewavelength of visible light. (Very clever work withoptical microscopes allows one to see down to about1/10th of the wavelength of visible light, to see objectslike microtubules.)

a) Incident and scattered wave together. b) After the incident wave has passed.

Figure 1If the scattering object is smaller than a wavelength,we get circular scattered waves that contain little orno information about the shape of the object.

incident wave incident wave

36-3

REFLECTION OF LIGHTUsing the picture of scattering provided by Figure (1),we can begin to understand the reflection of visiblelight from a smooth metal surface. Suppose we have along wavelength wave impinging on a metal surfacerepresented by a regular array of atoms, as illustrated inFigure (2). As the wave passes over the array of atoms,circular scattered waves emerge. As seen in Figure(2a), the scattered waves add up to produce a reflectedwave coming back out of the surface. The angleslabeled θi and θr in Figure (2b) are what are called theangle of incidence and angle of reflection , respec-tively. Since the scattered waves emerge at the samespeed as the incident wave enters, it is clear from thegeometry that the angle of incidence is equal to theangle of reflection. That is the main rule governing thereflection of light.

What happens inside the material depends upon detailsof the scattering process. Note that the reflectedwavefront inside the material coincides with the inci-dent wave. For a metal surface, the phases of thescattered waves are such that the reflected wave insidejust cancels the incident wave and there is no waveinside. All the radiation is reflected. For other types ofmaterial that are not opaque, the incident and scattered

waves do not cancel. Instead they add up to produce anew, transmitted wave whose crests move slower thanthe speed of light. This apparent slowing of the speedof light, due to the interference of transmitted andscattered waves, leads to the bending of a beam of lightas it enters or leaves a transparent medium. It is thisbending that allows one to construct lenses andoptical instruments.

Exercise 1

Using Figure (2), prove that the angle of incidenceequals the angle of reflection.

angle ofincidence

mirror

angle ofreflection

θi θr

Figure 2bWhen light reflects from a mirror, the angleof incidence equals the angle of reflection.

reflected wave

incident wave

angle ofincidence

angle ofreflection

θi θr

Figure 2aA reflected wave is produced when the incident wave isscattered by many atoms. From this diagram, you can seewhy the angle of incidence equals the angle of reflection.

36-4 Scattering of Waves

X RAY DIFFRACTIONIf the wavelength of the light striking a crystal becomescomparable to the spacing between atoms, we get anew effect. The scattered waves from adjacent atomsbegin to interfere with each other and we get diffractionpatterns.

The spacing between atoms in a crystal is of the orderof a few angstroms. (An angstrom, abbreviated A

o

, is 10– 8cm . An angstrom is essentially the diameter of a

hydrogen atom.) Light with this wavelength is in the xray region. Using Einstein's formula E = hf = hc/λ ,but in the form

E (in eV) = 12.4 × 10– 5 eV⋅cm

λ in cm

we see that photons with a wavelength of Ao

2 have anenergy

E photon with 2A°

wavelength = 12.4 × 10– 5eV⋅cm2 × 10– 8cm

= 6,200 eV (1)

This is a considerably greater energy than the 2 to 3 eVof visible photon.

When a beam of x rays is sent through a crystalstructure, the x rays will reflect from the planes ofatoms within the crystal. The process, called Braggreflection, is illustrated for the example of a cubiclattice in Figure (3). The dotted lines connect lines ofatoms, which are actually planes of atoms if youconsider the depth of the crystal. An incident wavecoming into the crystal can be reflected at variousangles by various planes, with the angle of incidenceequal to the angle of reflection in each case.

When the wavelength of the incident radiation iscomparable to the spacing between atoms, we get astrong reflected beam when the reflected waves fromone plane of atoms are an integral number of wave-lengths behind the reflected waves from the planeabove as illustrated in Figure (4). If it is an exactintegral wavelength, then the reflected light from all theparallel planes will interfere constructively giving usan intense reflected wave. If, instead, there is a slightmismatch, then light from relatively distant planes willcancel in pairs and we will not get constructive interfer-ence. The argument is similar to the one used to find themaxima in a diffraction grating.

incident X rays

reflected X rays

Planes of atoms

incident X rays

reflected X rays

Figure 3Planes of atoms act like mirrors reflecting X rays.

Figure 4When the incident X ray wavelength equals the spacingbetween one of the sets of planes, the reflected wavesadd up to produce a maxima.

36-5

Thus with Bragg reflection you get an intense reflec-tion only from planes of atoms, and only if the wave-length of the x ray is just right to produce the construc-tive interference described above. As a result, if yousend an x ray beam through a crystal, you get diffrac-tion pattern consisting of a series of dots surroundingthe central beam, like those seen in Figure (5). Figure(5a) is a sketch of the setup and (5b) the resultingdiffraction pattern for x rays passing through a silverbromide crystal whose structure is shown in (5c).

The main use of x ray diffraction has been to determinethe structure of crystals. From the location of the dotsin the x rays' diffraction photograph, and a knowledgeof the wavelength of the x rays, you can figure out theorientation of and spacing between the planes of atoms.By using various wavelength x rays, striking the crystalat different angles, it is possible to decipher complexcrystal structures. Figure (6) is one of many x raydiffraction photographs taken by J. C. Kendrew of acrystalline form of myoglobin. Kendrew used these xray diffraction pictures to determine the structure of themyoglobin molecule shown in Figure (17-3). Kendrewwas awarded the 1962 Nobel prize in chemistry for thiswork.

crystal

incident X rayreflected rays

film

Figure 5X ray diffraction study of a silver bromide crystal.

Figure 6One of the X ray diffraction photographs used byKendrew to determine the structure of theMyoglobin molecule.

c) The silverbromide crystal isa cubic array withalternating silverand bromineatoms.

b) X ray diffraction pattern produced by a silverbromide crystal. (Photograph courtesy of R. W.Christy.)

a) An incident beam of X rays is diffractedby the atoms of the crystal.

Figure 17-3The Myoglobin molecule, whose structurewas determined by X ray diffraction studies.

36-6 Scattering of Waves

Diffraction by Thin CrystalsThe diffraction of waves passing through relativelythin crystals can also be analyzed using the diffractiongrating concepts discussed in Chapter 33. Suppose forexample, we had a thin crystal consisting of a rectangu-lar array of atoms as shown in Figure (7a). The edgeview of the array is shown in (7b). Here each dotrepresents the end view of a line of atoms.

Now suppose a beam of waves is impinging upon thecrystal as indicated in Figure (7b). The impingingwaves will scatter from the lines of atoms, producingan array of circular waves as shown.

Compare this with Figure (8), a sketch of wavesemerging from a diffraction grating. The scatteredwaves from the lines of atoms, and the waves emerg-ing from the narrow slits have a similar structure andtherefore should produce similar diffraction patterns.

edge

view

lines ofatoms

incidentwave

lines of atoms

scattered waves

incidentwave

diffraction grating

emerging waves

Figure 7aFront view of a rectangular arrayof atoms in a thin crystal.

Figure 8The waves emerging from a diffraction grating have asimilar structure as waves scattered by a line of atoms.

Figure 7bEdge view with an incident wave. Each dot nowrepresents one of the line of atoms in Figure (7a).

alternatelines ofatoms

Figure 9Various lines of atoms can imitateslits in a diffraction grating.

Figure 10aA laser beam sent through asingle grating. The lines of thegrating were 25 microns wide,spaced 150 microns apart.

36-7

There is one major difference between the array ofatoms in Figure (7) and the diffraction grating of Figure(8). In the crystal structure there are numerous sets oflines of atoms, some of which are indicated in Figure(9). Each of these sets of lines of atoms should act asan independent diffraction grating, producing its owndiffraction pattern. The main sets of lines are horizon-tal and vertical, thus the main diffraction pattern weshould see should look like that produced by twodiffraction gratings crossed at right angles. Sending alaser beam through two crossed diffraction gratingsproduces the image shown in Figure (10). In Figure(10a), the laser beam is sent through a single grating. In(10b) we see the effect of adding another gratingcrossed at right angles.

Exercise 2In Figure (10a) the maxima seen in the photograph are1.68 cm apart and the distance from the grating to thescreen is 4.00 meters. The wavelength of the laserbeam is 6.3 × 10– 5cm. What is the spacing between theslits of the diffraction grating?

Exercise 3

In Figure (11), a laser beam is sent through two crosseddiffraction gratings of different spacing. Which image,(a) or (b) is oriented correctly? (What happens to thespacing of the maxima when you make the grating linescloser together?)

Figure 10bA laser beam sent throughcrossed diffraction gratings.Again the lines of the gratingwere 25 microns wide, spaced150 microns apart.

Figure 11Two diffraction gratings withdifferent spacing are crossed. Asshown, the vertical lines arefarther apart than the horizontalones. Which of the two images ofthe resulting diffraction patternhas the correct orientation?

a)

b)

36-8 Scattering of Waves

THE ELECTRONDIFFRACTION EXPERIMENTOne of the main differences between the scattering ofx rays and of electrons is that x ray photons interact lessstrongly with atoms, with the result that x rays canpenetrate deeply into matter. This enables doctors tophotograph through flesh to observe broken bones, orengineers to photograph through metal looking forhidden flaws. Electrons interact strongly with atoms,do not penetrate nearly as deeply, and therefore are wellsuited for the study of the structure of surfaces or thincrystals where you get considerable scattering from afew layers of atoms.

The Graphite CrystalGraphite makes an ideal substance to study by electronscattering because graphite crystals come in thin sheets.A graphite crystal consists of a series of planes ofcarbon atoms. Within one plane the atoms have thehexagonal structure shown in Figure (12), reminiscentof the tiles often seen on bathroom floors. The spacingbetween neighboring atoms in each hexagon is 1.42 A

o

as indicated at the bottom of Figure (12).

The atoms within a plane are very tightly boundtogether. The hexagonal array forms a very strongframework. The planes themselves are stacked on topof each other at the considerable distance of 3.63 A

o

asindicated in Figure (13). The forces between theseplanes are weak, allowing the planes to easily slide overeach other. The result is that graphite is a slipperysubstance, making an excellent dry lubricant. In con-trast, the strength within a plane makes graphite anexcellent strengthening agent for epoxy. The resultingcarbon filament epoxies, used for constructing racingboat hulls, light airplanes and stayless sailboat masts, isone of the strongest plastics available.

d1

d =2.13A

1

1.42A

effective gratings

o

o

Figure 12The hexagonal array of atoms in one layer of agraphite crystal. Lines of atoms in this crystalact as crossed diffraction gratings.

planeseparation= 3.63A

o

Figure 13Edge view of the graphite crystal, showing theplanes of atoms. The planes can easily slide overeach other, making the substance slippery.

36-9

The Electron Diffraction TubeThe electron diffraction experiment where we sent abeam of electrons through a graphite crystal, can beviewed either as an experiment to demonstrate thewave nature of electrons or as an experiment to studythe structure of a graphite crystal. Perhaps both.

The apparatus, shown in Figure (14), consists of anevacuated tube with an electron gun at one end, agraphite target in the middle, and a phosphor screen atthe other end. A finely collimated electron beam can beaimed to strike an individual flake of graphite, produc-ing a single crystal diffraction pattern on the phosphorscreen. Usually you hit more than one crystal and geta multiple image on the screen, but with some adjust-ment you can usually obtain a single crystal image.

Electron WavelengthThe accelerating voltage required to produce a gooddiffraction pattern is in the range of 6,000 volts. As ourfirst step in the analysis, let us use the de Brogliewavelength formula to calculate the wavelength of6,000 eV electrons.

The rest energy of an electron is .51 MeV, or 510,000eV, far greater than the 6,000 eV we are using in thisexperiment. Since the 6,000 eV kinetic energy is muchless than the rest energy, we can use the nonrelativisticformula 1/2 mv2 for kinetic energy. First converting6,000 eV to ergs, we can equate that to 1/2 mv2 tocalculate the speed v of the electron. We get

6000 eV × 1.6 × 10– 12ergseV = 1/2 mev

2 (2)

With the electron mass me = .911 × 10– 27gm , we get

v2 =

2 × 6000 × 1.6 × 10– 12ergs

.911 × 10– 27gm

= 21.1 × 1018cm2

sec2

v = 4.59 × 109cm/sec (3)

which is slightly greater than 10% the speed of light.

The next step is to calculate the momentum of theelectron for use in de Broglie's formula. We have

p = mv

= .911 × 10– 27gm × 4.59 × 109cmsec

= 4.18 × 10– 18gm cmsec

(4)

Finally using de Broglie's formula we have

λ = h

p =6.63 × 10– 27gm cm2/sec4.18 × 10– 18gm cm/sec

λelectron = 1.59 × 10– 9cm = .159 °A (5)

Thus the wavelength of the electrons we are using inthis experiment is about one tenth the spacing betweenatoms in the hexagonal array.

Exercise 4

Calculate the wavelength of a 6000 eV photon. Whatwould cause such a difference in the wavelengths of aphoton and an electron of the same energy?

graphite crystalphosphorscreen

electron gun

electron beam

diffractedelectrons

18 cm

Figure 14Electron diffraction apparatus. An electron beam,produced by an electron gun, strikes a graphitecrystal located near the center of the evacuated tube.The original beam and the scattered electrons strikea phosphor screen located at the end of the tube.

36-10 Scattering of Waves

The Diffraction PatternWhat should we see when a beam of waves is diffractedby the hexagonal array of atoms in a graphite crystal?Looking back at the drawing of the graphite crystal,Figure (12), we see that there are prominent sets of linesof atoms in the hexagonal array. To make an effectivediffraction grating, the lines of atoms have to be equallyspaced. We have marked three sets of equally-spacedlines of atoms, each set being at an angle of 60° from eachother. We expect that these lines of atoms shouldproduce a diffraction pattern similar to three crosseddiffraction gratings.

In Figure (15), we are looking at the diffraction we getwhen a laser beam is sent through three crossed diffrac-tion gratings. In (15a), we have 1 diffraction grating. In(15b) a second grating at an angle of 60° has been added.In (15c) we have all three gratings, and see a hexagonalarray of dots surrounding the central beam, the centralmaximum.

Figure (16) is the electron diffraction pattern photo-graphed from the face of the electron diffraction tubeshown in Figure (14). We clearly see an hexagonal arrayof dots expected from our diffraction grating analysis.On the photograph we have superimposed a centimeterscale so that measurements may be made from thisphotograph.

0 1 2 3 4 cm

d1

effective gratings

Figure 15aSingle gratingdiffractionpattern.

Figure 15bTwo gratingdiffractionpattern.

Figure 15cDiffractionpattern fromthree crossedgratings.

Figure 12 (section)Three sets of lines of atoms act as three crosseddiffraction gratings with 2.13 angstrom spacing.

Figure 16Diffraction pattern produced by a beam of electronspassing through a single graphite crystal. The energyof the electrons was 6000 eV.

d =2.13A

1

1.42A

o

o

36-11

The electron diffraction apparatus allows us to movethe beam around, so that we can hit different parts of thetarget. In Figure (16), we have essentially hit a singlecrystal. When the electron beam strikes several graph-ite crystals at the same time, we get the more complexpattern seen in Figure (17).

Analysis of the Diffraction PatternLet us begin our analysis of the diffraction pattern byselecting one set of dots in the pattern that would beproduced by one set of lines of atoms in the crystal. Thedots and the corresponding lines of atoms are shown inFigure (18). In (18a) we see that the spacing Ymaxbetween the dots on the screen is 1.33 cm. Thesehorizontal dots correspond to the maxima for a verticalset of lines of atoms indicated in (18c). In (18b) we arereminded that the distance from the target to the screenis 18 cm. Using the diffraction grating formula, we cancalculate the wavelength of the electron waves thatproduce this set of maxima.

Figure 17Diffraction pattern produced by a beam of electronspassing through multiple graphite crystals.

Using the diffraction formula, Equation 33-3, andnoting that Ymax << D, we have

λ = Ymaxd

D2 + Ymax2

≈ YmaxdD

= 1.33 cm × 2.13 × 10– 8cm18 cm

λ = 1.57 × 10– 9cm (6)

which agrees well with Equation 5, the calculation ofthe electron wavelength using the de Broglie wave-length formula.

d1

d =2.13 A

1 o

graphite crystal

diffractiongratingmaxima

diffractedelectrons

18 cm

Figure 18aThe diffraction grating maxima from one set of lines inthe graphite crystal. You can see that 3ymax = 4cm, sothat ymax = 1.33 cm.

Figure 18bTop view of the electron diffraction apparatus..

Figure 18cThe vertical lines of atoms in the graphite crystal thatproduce the horizontal row of dots seen in (a).

36-12 Scattering of Waves

Other Sets of LinesWith a careful analysis of the lines of atoms in thehexagonal ray of atoms, one can explain all the dots ofthe diffraction pattern of Figure (16). For example, inFigure (19) we see that there is another set of lines thatare rotated at an angle of 30° and more closely spacedthan our original set. In Figure (20), we have high-lighted a set of dots in the diffraction pattern that arerotated by an angle of 30° and more widely spaced thanthe dots we have been analyzing. Since more closelyspaced lines in a grating produce more widely spacedmaxima, we should suspect that the highlighted maximaresult from this new set of lines. The point of Exercise5 is to see if this is true.

Exercise 5(a) Explain why more closely spaced atoms shouldproduce more widely spaced dots in the diffractionpattern.

(b) Assuming that the dots highlighted in Figure (20) areproduced by the lines of atoms shown by dotted lines inset 2 of the effective gratings, calculate the wavelengthof the waves producing the dots. Compare your resultswith our previous analysis.

Exercise 6

Suppose that a beam of neutrons rather than electronswere fired at the graphite crystal. Assuming that neu-trons also obey the de Broglie relationship λ = h/p, whatshould be the kinetic energy, in eV, of the neutrons inorder to produce the same diffraction pattern with thesame spacing between dots?

d1

d =2.13A

1

d2

d =1.22A

1.42A

2

effective gratingsset 1

effective gratingsset 2

o

o

o

0 1 2 3 4 cm

Figure 19It is easy to find a second set of effectivegratings, rotated 30° from the first set,and with a narrower spacing.

Figure 20We have highlighted themaxima produced by thisset of lines. Note that themore narrowly spacedlines produce morewidely spaced maxima.

36-13

Figure 21Microphotograph of the three crossed diffractiongratings. The lines are 25 microns wide and 100microns apart, on centers. (Student project by BradyBeale and Amy Coughlin.)

Figure 15cDiffraction pattern produced by a laser beam goingthrough the three crossed gratings of Figure 21.

Figure 22Microphotograph of a hexagonal dot pattern. The dotsare 25 microns in diameter and 100 microns apart.(Student project by Brady Beale and Amy Coughlin.)

Figure 23Diffraction pattern produced by a laser beam goingthrough the hexagonal dot pattern of Figure 22.

Student ProjectsThe crossed diffraction gratings used to obtain thevarious laser diffraction patterns in this chapter, werecreated using the Adobe Illustrator program, and thenprinted on film using a Linatronic imagesetter at a localdesktop publishing company. The one micron resolu-tion of the imagesetter allowed us to construct variousgrating and dot patterns that produced reasonablediffraction patterns with a laser.

Several students doing project work with these gratingsand dot patterns suspected that some patterns were notas good as they should be and took microscope photo-graphs of them. They found that lines or dots as smallas 10 microns wide tended to be filled in and blotchy,but lines or dots 25 microns wide came out fairly wellas can be seen in Figures (21) and (22). Figure (23) isthe laser diffraction pattern produced by a laser beampassing through the hexagonal dot pattern of Figure(22).

36-14 Scattering of Waves

incidentwave

lines of atoms

scattered waves

incidentwave

diffraction grating

emerging waves

Student project by Gwendylin ChenIn our discussion of the diffraction of waves by theatoms of a crystal, we pointed out that waves shouldemerge from a line of atoms in much the same wah thatthey do from the slits of a diffraction grating. The twosituations were illustrated in Figures (7b and 8) repro-duced below.

That a slit and a line produce similar diffraction pat-terns was clearly illustrated in a project by GwendylinChen. While working with a laser, she observed thatwhen the beam passed over a strand of hair it produceda single slit diffraction pattern superimposed on theimage of the beam itself. Here we have reproducedGwendylin’s experiment. Figure (24) is a photographof a slit made from two scapel blades, and a strand ofGwendylin’s hair. We tried to make the width of the slitthe same as the width of the hair. The two circlesindicate where we aimed the laser for the two diffrac-tion patterns.

The results are seen in Figure (25).The diffractionpatterns are almost identical. The only difference isthat when the beam passes over the hair, it continues onlanding in the center of the diffraction pattern.

a) single slit diffraction pattern

b) diffraction pattern produced by strand of hair

Figure 8The waves emerging froma diffraction grating havea structure similar to thewaves scattered by a line ofatoms.

Figure 7bEdge view of a thincrystal with an incidentwave. Each dot nowrepresents one of the lineof atoms in the crystal.

Figure 24Slit and hair used to produce diffractiopn patterns.The circles indicate where we aimed the laser.

Figure 25Comparason of diffeaction patterns.

incident laser beam

strand of hair

slit formed by two scapels

laser through slit

laser past hair

36-15

IndexAAngle of reflection (scattering of light) 36-3

BBragg reflection 36-4

CCarbon

Graphite crystal, electron diffraction 36-8Crystal

Diffraction by Thin 36-6Graphite, electron diffraction by 36-8Structures

Graphite 36-8X ray diffraction 36-5

DDiffraction

By thin crystals 36-6Electron diffraction tube 36-9X Ray 36-4

Diffraction, electron. See Experiments II: -11- Electrondiffraction experiment

Diffraction patternAnalysis of 36-11By strand of hair 36-14Electron 36-10For x rays 36-5Of human hair 36-14Student projects 36-13

EElectron

Diffraction Pattern 36-10Electron diffraction experiment 36-8

Diffraction tube 36-9X. See Experiments II: -11- Electron diffraction

experimentElectron scattering

Chapter on scattering 36-1Electron waves

Wavelength of 36-9Energy

Kinetic energyElectron diffraction apparatus 36-9

X Ray photons, energy of 36-4Experiments II

-11- Electron diffraction experiment 36-8

GGraphite crystal

Electron diffraction experiment 36-8Electron scattering 36-1Structure of 36-8

HHair, strand of, diffraction pattern of 36-14Hexagonal array

Graphite crystal and diffraction pattern 36-9

KKinetic energy

Electron diffraction apparatus 36-9

LLenses, transmitted waves 36-3Light

Diffraction of lightBy thin crystals 36-6Pattern, by strand of hair 36-14Patterns, student projects 36-13

Reflection 36-3X ray diffraction 36-4

MMeiners, Harry, electron scattering apparatus 36-1

PParticle-wave nature

Of electronsElectron diffraction experiment 36-8

RReflection

Bragg reflection 36-4Of light 36-3

SScattering of waves

By graphite crystal, electron waves 36-8By myoglobin molecule 36-5By small object 36-2By thin crystals 36-6Chapter on 36-1Reflection of light 36-3X ray diffraction 36-4

TTransmitted wave and lenses 36-3

WWave

Transmitted waves 36-3Wavelength

Electron 36-9

Xx-Ch36

Exercise 1 36-3

36-16 Scattering of Waves

Exercise 2 36-7Exercise 3 36-7Exercise 4 36-9Exercise 5 36-12Exercise 6 36-12

X-raysDiffraction 36-4Diffraction pattern 36-5Photon energies 36-4