Xray Diffraction Report 3r4b3g

Experimentally Determining the Molecular Separation Distance in Sodium Chloride via X-Ray Diffraction

Created By Brian Hallee

Partnered by Joseph Oxenham

Performed November 6, 2010

Historical Background X-Rays officially entered into the realm of human understanding by way of Wilhelm Conrad Rontgen’s discovery of them in 1895. As a professor of Physics at the University of Wurzburg, he had previously performed experiments on cathode rays to determine their photographic properties. In his famous hour, a lonely Friday afternoon in a dark room in the physics building, Rontgen was utilizing a “Crookes-tube”, (a vacuumed tube with a positive and negative electrode at either end), to focus cathode rays towards a florescent screen coated with barium platinocyanide.1 The Crookes tube was known to give off a glow at the edges when in use. Thus, Rontgen buffered the glow from the screen he was observing by wrapping the tube in black cardboard. To his fascination, the screen continued to display a faint glow of which, due to the relatively large distance from the tube, the source could not possibly have been stray cathode rays. In reality, “hard” X-rays, with their solid penetrating abilities, were being created via Rontgen’s apparatus. Rontgen seemed to quickly grasp the significance in application of these rays, as he made popular his findings by performing the first radiological X-Ray scan of his wife’s hand2 (See Figure 1). The discovery immediately thrust Rontgen into the limelight of the physics world, won http://en.wikipedia.org/wik i/File:Anna_Berthe_Roentg en.gif

him the first Nobel Prize in Physics ever bestowed3 (1901), and paved the way for countless applications Page | 1

of the x-rays wondrous abilities such as the topic of this paper. The notion that X-rays could determine the spacing of atoms in salt came after a series of quick “bursts” of findings over just a few short years. The ball began rolling when the German physicists Max von Laue and Paul Ewald conversed in the summer of 1912 over Ewald’s thesis experiment.4 The idea was to use crystals as diffraction gratings for visible light. However, von Laue quickly realized the model fell short due to the molecular spacing in the crystal being of shorter distance than the wavelength of light. Thus, the spacing becomes physically impossible to resolve using the visible spectrum of electromagnetic radiation. Von Laue proposed the use of X-rays for this task, as their wavelength falls below the range of molecular distances (10 – 0.01 nm)5. He went to work acquiring technicians and an X-ray source to aim a beam of rays through a copper sulphate crystal, and observe the diffraction on a specialized photographic surface. What he observed was a structured order of interference patterns arranged in circles around the spot produced by the direct beam. This made obvious two facts: X-rays are waves and the atoms in a crystal are arranged in a regular pattern. Following this and many other experiments, von Laue applied mathematical rigor to the phenomenon and derived physical laws governing the deflection angles and spacing of the patterns that result from X-ray diffraction. This work earned him the 1914 Nobel Prize in physics. Continuing the breakneck-pace at which the field was developing, and naturally inspired by von Laue’s genius and success, the father and son team of W.H. and W.L. Bragg sought to apply Page | 2

mathematics to the reflection of X-rays in the evenly-spaced planes of crystals. While we will leave the actual derivation to the theory section, as it directly applies to our sought-after results for this experiment, it is important to note that their conclusion, (now bestowed the term Bragg’s Law), gives a conscious experimenter the ability to measure the molecular spacing in any crystal, so long as the wavelength of the X-ray is pre-determined as well as the angle at which it arrives (relative to the face).6 The potential for application of this law was huge. Immediately following the discovery, teams of researchers began determining the structures and bonds present in the simplest of inorganic materials, and eventually working up to large organic chains. The first crystal to be attacked and solved was ordinary table salt, (Sodium Chloride, the material used in our own experiment). Once thought to be held by some form of covalent bond, the diffraction study effectively predicted the presence of ionic bonds.7 Owing to its monumental application, Bragg’s Law secured both W.L. and W.H. Bragg the Nobel Prize in physics for 1915.

Theoretical Basis Considering this entire experiment is closely coupled to the act of Xray generation, it is only fitting to briefly describe how this is acquired. In very simple , the “X-ray tube” we will utilize acts similarly to a cathode ray tube in that an electron is accelerated in some way. However, residing in the tube is a copper leaf that is in charge of actually generating the X-rays.

Figure scattering Figure 4: 3: X-Ray A model of the occurring on two Figure 2: Depicting X-Rays being geometry separate planes, and the associated Sodium and Chloride ions scattered in table saltfrom two planes of a crystal. Deriv.gif Source: See8

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Copper is chosen for the reason that it is known to behave in a certain way when injected with a high-speed electron. When the electron enters the copper nucleus, the atom emits a wide array of electromagnetic waves coined “Bremsstrahlung Radiation”. Immediately following this, a valence electron is emitted leaving the atom in an unstable state. At this point, a higher-energy electron will make a quantum jump to quell this issue while simultaneously emitting an Xray. When this phenomenon is compounded by a stream of steady high-speed electrons, a relatively steady stream of X-rays is also produced. As we mentioned earlier, in order to diffract, reflect, or resolve at such infinitesimally small distances such as the spacing of a molecule, the wavelength of the ray doing the job must be of similar order. Considering this experiment has been performed many times over, it was pre-determined that the accepted value for Sodium-Chloride crystal spacing is roughly 0.282nm.8 Thus, “hard” X-rays are required for this task as their wavelengths span from .1nm to 0.01nm. It is also known that when an atom accepts an X-ray at some angle relative to its plane (Grazing angle) that it will reflect it back at the same Grazing angle in the same direction it was moving prior to acceptance. This is called X-ray scattering, and it is depicted in figure 2. The figure denotes the accepting grazing angle as θa and the reflection angle as θr. As we will see later, we want all the rays Page | 2

scattered off of the same plane to be in phase. Thus, θa will equal θr. Also, in our experiment, Sodium Chloride exhibits equal spacing between atoms in a grid-like fashion (Figure 3). Thus, in reference to figure 2, both variable-less dimensions shown are actually equal to each other. This will aid us in determining the phasedifference of rays that scatter off planes directly above or below each other. One remaining important fact that applies to our following derivation is that our X-ray tube configuration emits two specific X-range wavelengths of 0.154nm and 0.138nm. We will denote the wavelengths as λα and λβ respectively, and λα is of greater intensity than that of λβ. Now, unlike a diffraction grating where all the emitted waves are in phase, X-rays scattered from crystals do not exhibit this uniformity. As you can plainly see from figure 2, the X-rays that are scattered from the 2nd layer of molecules travel a certain distance farther than those scattered from the first layer. Depending on this added distance, the lower waves may constructively or destructively interfere with the higher ones. We can determine exactly what distances would contribute to these phenomena by applying a bit of trigonometry to figure 2, as shown in figure 4. We can infer Page | 3

from figure 4 that the second ray must travel an added distance of AB + BC in order to remain parallel to the top ray once scattered. For to waves to be in phase, the “phase difference”, (distance from peak-to-peak or trough-totrough), must be an integer number times the wavelength of the waves in question. Thus, if we wish to assume our two waves in figure 4 are in phase, the added distance for the bottom ray must be equal to n*λ, or: nλ=AB+BC

(eqn. 1)

We may also infer from figure 4 that either AB or BC forms a mini triangle with hypotenuse d. Thus, sinθ= oppositehypotenuse = ABd → AB=d*sinθ

(eqn. 2)

Because Sodium Chloride exhibits equal spacing in both the vertical and horizontal directions, AB = BC, and we can rewrite equation 1 as follows: nλ=2AB

(eqn. 3)

All that is left is to substitute equation 2 into equation 3 and we achieve: 2d*sinθ = nλ

(eqn. 4 - Bragg’s Law)

N = An integer number (1, 2, 3, 4, …) d = The distance from atom-to-atom inside the molecule λ = The wavelength of the X-Ray used during the scattering θ = The Grazing Angle of the X-ray (Relative to the face of the crystal)

This is the original Bragg’s law. While the law is mathematically easy to derive, recall from before how this law single-handedly has allowed for any crystal’s structure to be determined if so desired. Notice that the location of Page | 4

the crystal does not come into play determining how the X-rays are scattered. So long as the crystal lies in the beam of rays, the phenomenon can be observed. Obviously, we started with an, at this point, seemingly reckless assumption that lengths AB + BC should place the 2nd ray in phase with the first. This is not always the case, and the entirety of this experiment is a rigorous attempt at finding exactly when equation 4 does apply in a real scattering scenario. Naturally, equation 4 is only relevant when X-rays are experimentally determined to be reflecting in phase from the crystal’s planes. Thus, a full sweep of angles needs to be done to observe the “key” angle (or angles) at which Bragg’s law applies. If equation 4 is applicable at a certain angle, only then we are able to solve for the spacing between molecules in the crystal.

Apparatus The centerpiece of this experiment was the Tel-X-Ometer Tel 580 machine. This device came equipped with the X-ray tube and a cord to power the tube. The tube was situated at the edge of a stage facing the center. A knurled clutch plate resided at the center that interlocks both the crystal post, (Where the sodium chloride is mounted), and the carriage arm. The carriage arm extends outward and off of the stage so the may adjust the angle at which the crystal receives the X-rays. The spectrometer drive mechanism driving the carriage is 2:1. Thus, for every degree change of the carriage the grazing angle changes by ½ degree. Containing all

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devices on the stage and shielding the from stray X-rays is a large lead/glass shield. The shield deactivates crucial switches when lifted up that effectively bar the X-ray tube from galvanizing in any fashion. Located directly opposite of the X-ray tube, attached to the shield, was a lead/aluminum plate placed for added protection against stray X-rays. Once the shield is down and in place, the machine is activated via the turning of a key and setting of a timer. A specified X-ray generation time must be set in order for the X-rays to begin emanating. In order to determine when we have arrived at a “Bragg-applicable” angle, we need some way at numerically observing the scattering of the rays. Thus, a Geiger Muller Tube is attached to the carriage for this purpose. An X-ray travels at the speed of light. Thus, as it enters the Geiger tube it will literally rip the electrons from the air molecules inside. A positively charged Tungsten wire resides inside to attract these electrons which acquire huge velocities of their own. More electrons will subsequently be freed thus creating a short surge in the wire allowing the Geiger tube to transmit data on the X-ray intensity. In order to achieve clean results, (and minimize stray X-rays), a series of collimators are used to narrow the beam of X-rays after leaving the X-ray tube, and before entering the Geiger tube. The collimators are lead disks that are slide mounted and come equipped with specified diameter slots (1mm – 3mm) for narrowing the beam. The primary collimator makes use of

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an O-ring to ensure that movement will not occur as the collimator heats up. Also positioned on the carriage arm was a thin slide of nickel film to filter out much of the λβ X-rays. We made use of two devices in order to gain inclination as to where the peaks, or “Bragg-applicable” angles, were. First, we made use of a speaker-equipped Geiger counter that connected directly to the Geiger Muller tube. This can be thought of as the “coarse” measurement tool, as it did not come equipped with any precise measuring tools to quantitatively handle the X-ray intensity. Its usefulness came in performing an initial run to determine roughly where the peaks occurred. The other, and more precise, device at our disposal was the ratemeter. This meter, when connected to the Geiger tube, was able to give us a numerical value to the number of Xray occurrences over a -specified interval. While the occurrences were, inherently, not of any importance, the angle at which they were greatest signified the angle at which we could safely apply Bragg’s Law.

Procedure Fortunately, before we even began the experiment most of the finer details of the apparatus were already in place. The Sodium Chloride crystal was pre-mounted into the crystal plate. The collimators and filters were already slide-mounted in their proper place. The Geiger Muller tube was properly mounted onto the carriage arm, and its signal was sent through wire to the Geiger counter and ratemeter. Specifically, the ratemeter was Page | 2

set to a potential of 420 Volts and the X-ray counting duration was set to 10 seconds. Therefore, our intensity for the experiment would be in units of occurrences per 10 seconds. To begin the lab, our job was to simply ensure the devices were calibrated correctly, and to perform an initial test run. The final steps taken before activating the X-rays were included ensuring the TelX-Ometer Tel 580 machine was set to 30kV, turning the key to activate the machine, and setting the timer to 50 minutes. At this point, with the shield down and locked in place, we were able to begin the scattering process. As mentioned, we performed a quick run-through of all the allowable angles on the machine (roughly 12 to 120 degrees) with the Geiger tube connected to the Geiger counter. In listening carefully to the intensity of the sound emanating from the Geiger counter, we were able to roughly decide where the peaks occurred to a certainty of about one degree. This gave us an idea as to where we should “home in” on a peak when using the more quantitative ratemeter. However, we were only able to determine roughly where the first four peaks occurred (28, 31, 56, and 59 carriage degrees respectively). Consequently, the others had to be found with the ratemeter. Thus, following our initial run, we connected the Geiger Muller tube to the ratemeter and set the carriage back to 12 degrees. This time, we moved the carriage in increments of one degree on the machine (1/2 a degree of Grazing angle!) and stopped to take a 10 second measurement at each angle. The ratemeter was extremely -friendly, and only required the push of a button at each angle to perform a ten second measurement of the Page | 2

amount of X-rays present in the tube. At the end of the measurement, the display on the ratemeter would read the exact number of occurrences in similar fashion to that of a digital alarm clock. We took intensity measurements in this fashion until reaching 120 degrees on the machine. At this point, we utilized the approximate-peak angles we found in our test run and traveled back to them performing intensity measurements at every 1/6th of a degree. At the very least, we attempted to perform these fine measurements up to a degree before and after the aforementioned peaks. All of the angles and their associated intensities were simultaneously being plotted in a Microsoft Excel spreadsheet. Thus, acquiring a large amount of data points around the peaks enabled us to construct a very vivid and defined peak in our graph. As mentioned in the theory section of this report, the particular X-ray tube used in our experiment generated two distinct wavelengths in the X-ray range. Due to the already defined value of molecular spacing in our sample, we expect both of these wavelengths to exhibit three different peaks each in the interval of angles we are working with. Thus, we expected to find a total of six peaks in our graph at the conclusion of the experiment. The final two peaks, (corresponding to n = 3 for λα and λβ), are extremely small and undefined. Therefore, it was not expected that we necessarily determine (even roughly) where these peaks occurred during the course of our initial test run. Instead, we determined the whereabouts of these peaks by looking at our graph after performing a quantitative measurement at every 1 degree. Due to certain inexplicable Page | 3

scatterings and background radiation, both the Gieger counter and the rate meter always picked up some amount of X-rays at virtually every angle. This further demonstrates the importance of having numerical values to look at in determining when most of those X-rays measured were due to the pertinent effect of scattering. After we determined to some acceptable degree of precision where our peaks occurred, the X-ray machine was keyed off, and both the Geiger counter and ratemeter were powered down. In order to find the separation distances in our sample of Sodium Chloride, the calculation could be performed at home after gathering our Bragg-applicable angles from the lab experiment.

Sample Calculations While this lab remains relatively void of any rigorous mathematical work to achieve a solution, there are a few calculations that are pertinent to achieving the final value for separation distance and understanding its error. This section will give a brief overview of all the calculations used to generate the raw data and graph (See Appendix and Discussion sections), and how we will determine such properties such as relative error and standard deviation to be discussed in the following section. First, we have mentioned that all the angles marked on the Tel-X-Ometer stage are actually twice the grazing angle pertinent to our calculations. Thus, to generate the grazing angles found in our raw data, we used the following formula: Page | 4

Grazing Angle= Stage Angle2

(1)

Therefore, using (1), if we are measuring the X-ray intensity at an angle of 65 degrees on the stage, the actual grazing angle can be found as follows: Grazing Angle= 65°2= 32.5°

This is truly the only mathematical operation needed to generate the graph (see Appendix – Disc Contents or Figure 5) we did. After forming a 2-column list of Grazing angle vs. our intensity measurements we generated the graph in Excel using the following formula: Graph Data =Sheet1!$B$4:$C$159

After forming the graph and determining exactly where our peak lies, we can finally apply Bragg’s law (equation 4) to our data and determine and experimental value for molecular separation: nλ = 2d*sin⁡(θpeak)

(2)

We are ready to apply (2) to a peak to demonstrate a sample calculation. Looking at figure 5 we will take the first λα peak into and apply Bragg’s law to it as follows: 1*1.54x10-10m=2*d*sin15.583° → d= 1.54x10-10m2*sin15.583° dα1=2.866x10-10m ≈ 0.286nm

Figure 5: X-ray Intensity graph generated during the lab procedure. (All angles Page | 2 are actual grazing angles)

We mentioned previously that the actual separation distance for NaCl is 2.82nm, so this is very close. We will quickly use the same method (2) to find the d-value for the first λβ peak, and then average the two d-values. dβ1= 1.38x10-10m2*sin13.998° = 2.853x10-10 = 0.285nm

In the following discussion section, we will take a careful look at all of our dvalues and their associated errors and deviations. Thus, we will present here a brief summary as to how these calculations are performed. As we will see, the average of our values will come in hand for certain calculations, so we quickly determine this using the following formula: Average= d = 0ndnn

(3)

In (3), n represents the total number of data points. Continuing with our sample calculations, we will use the determined values of dα1and dβ1 and find their average as follows: d1 = 0.285nm+0.286nm2 = 0.2855 nm

This value is of importance in solving for a statistical term known as the standard deviation. The standard deviation, denoted by σ, is found via the following formula: σ= 0n(dn- d)2n

(4)

This is important in observing how our data varies, and we achieve a sample deviation using the two previous values and their average below:

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σ = (0.286nm-0.2855nm)2+ (0.285nm-0.2855nm)22 = 0.0005

There are a few other important calculations relevant to finding the error of our values, and we will run through them quickly. First, the absolute error of a value relative to its accepted value is found via: Abs. Error=| dreal - dexperiment |

(5)

Therefore, using (5), the absolute error of our first alpha peak is as follows: .282nm- .286nm = .004nm

We can, and will, use this notion of absolute error to find to important error values: Relative and percent-Relative error. They are respectively calculated as follows: Rel. Error= Abs. Errordreal % Rel. Error = Rel. Error*100%

(6) (7)

Using our previous values, we will quickly find some sample errors below:

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Rel. Error = 0.004nm0.282nm = 0.0142 % Rel. Error = (0.0142) * 100% = 1.42% error

This is not too shabby for a quick look at two peaks, and we can now feel content that both our data and calculations are accurate and can be discussed and applied to theory.

Discussion Looking at our graph (Figure 5), we feel as though we were able to acquire reasonably defined peaks acceptable for the application of Bragg’s Law. Even if we did not, the notion that the Tel-X-Ometer carriage would be almost impossible to rotate at angles smaller than 1/6th of a degree means that our data is likely as good as we could possible achieve with such equipment at our disposal. Using the Grazing angles associated with the highest peaks in our data (See Appendix – Raw Data) we can achieve our six values for d by applying equation 4 to the angles. ( that the β peaks are smaller and come before the α’s. Likewise, keep in mind that every peak we increasingly come across that corresponds to the same wavelength (λα or λβ) will require an integer jump in n for Bragg’s Law to apply). dα1 = 1*0.154x10-9m2*sin15.583° = 2.86635x10-10m = 0.2867nm dβ1 = 1*0.138x10-9m2*sin13.998° = 2.85256x10-10m = 0.2853nm dα2 = 2*0.154x10-9m2*sin33° = 2.82756x10-10m = 0.2828nm

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dβ2 = 2*0.138x10-9m2*sin29.32° = 2.81813x10-10m = 0.2818nm dα3 = 3*0.154x10-9m2*sin(54.391°) = 2.8413x10-10m = 0.2841nm dβ3 = 3*0.138x10-9m2*sin(47.5°) = 2.80763x10-10m = 0.2808nm

The six values for d are relatively stable throughout the peaks. We can numerically derive the “jitter” in our values by calculating the standard deviation of our values using (4). Omitting the units, σ = 0.2867-0.2822+ 0.2853-0.2822+ 0.2828-0.2822+ 0.2818-0.2822+ 0.28410.2822+ 0.2808-0.28226*10-9 = 2.566 x10-12

Therefore, we had a mean error or variance of about 0.003 nm which, in our case, is lightly significant but not detrimental. In order to find a solid value for d, we will average the six values together using (3) shown below: Avg= d = 0.2867nm+0.2853nm+0.2828nm+0.2818nm+0.2841nm+0.2808nm6 = 0.2836nm

Next, we wish to achieve the percent relative error against the accepted value for d and express what might have caused such error in our experiment. We start by calculating the absolute error relative to the accepted value of 0.282nm for NaCl using (5): Abs. Error=0.282nm- 0.2836nm= 0.00158nm

We are now in position to find the % error relative to the accepted value using (6) and (7):

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% Rel. Error = 0.00158nm0.282nm*100 % = 0.5615% error

While, holistically, this is not a decent amount of error in any calculation, there are a few factors that may have inevitably led to it. As you can infer from our graph (Figure 5), the peaks at the n=3 range were barely even noticeable (especially the final λβ peak). We also lightly mentioned that the Tel-X-ometer only enabled us to measure to a precision of 1/6th of a degree. Consequently, it was not possible to fully home in on where the peak lied when n=3 due to the limitations of the carriage and the barely noticeable peaks at this range. This may have contributed to some of the error garnered via our last two measurements. A carriage with a higher degree of accuracy would have enabled us to further home-in on a more accurate peak. Another possible source of error was the relatively large amount of “noise” or background radiation in all of our measurements. The Tel-X-Ometer is, at the least, decades old and may exhibit X-ray leakage of some-sort throughout the stage. Nonetheless, every single measurement we took exhibited intensity (X-rays per 10 seconds) in the high-100’s or low-200’s. Some of this variable background noise may have contributed to a possible skewing of a peak, thereby introducing some error in our values. Nevertheless, an experimental value with a relative error below one percent is a very successful run by any standards. This is especially true considering the distances at which we are measuring, and the way in which we are indirectly measuring them. As we have mentioned many times, physics textbooks8 state that the accepted value of d is less than one nanometer. Page | 2

This distance is much smaller than the smallest semi-conductor that can be manufactured today by the most precise and sophisticated manufacturing equipment. Therefore, to exhibit an absolute error of less than a hundredth of a nanometer by observing peaks of X-rays is simply extraordinary. Although we previously mentioned that d was the distance separating the molecules in NaCl both vertically and horizontally, this paper has, thus far, assumed that you have simply treated that fact as an axiom. However, we are now in position to prove that this, in fact, must be the case. Introductory electromagnetism suggests that if all of the atoms in a substance were not evenly spaced and ordered, then a beam of X-rays aimed at the substance would scatter in every direction constructively and destructively interfering in a seemingly random fashion. However, we have proved through our graph (Figure 5) that at certain angles all of the X-rays will scatter and constructively interfere, suggesting that Sodium and Chloride atoms are arranged in a cubic fashion in the substance. This is the very definition of a crystal. Therefore, we have proved, (likely for the 100-millionth time), that table salt is a crystal. In conclusion, we have effectively predicted through this experiment (using Bragg’s Law) that table salt forms a crystal lattice and that lattice is measured in nanometers (Or Angstroms, if you prefer). Because of these infinitesimally small distances, we are required to measure the atomicseparation indirectly by use of electromagnetic radiation. This is due to the

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fact that visible light has wavelengths in the hundreds of nanometers, effectively barring us from visually resolving the separation. This also bars us from using visible or even ultraviolet light to scatter off the crystal, as they will not be effectively absorbed and re-emitted. Therefore, X-rays are the tool of choice and their scattering off the crystal at certain angles gives us the ability to apply Bragg’s law and solve for the distance. Miraculously, as we saw via the derivation of Bragg’s Law (see Theoretical Basis), mathematical proof of this lies only in simple algebra and light trigonometry. Therefore, through this experiment, X-ray diffraction has shown to be perhaps the most simple and powerful tool available in cracking the code behind crystal structures.

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Works Cited Hugh D. Young, R. A. (2007). University Physics. Pearson Addison-Wesley. Laue, M. v. (1915, November 12). Concerning the Detection of X-Ray Interferences. Retrieved November 10, 2010, from NobelPrize.org: http://nobelprize.org/nobel_prizes/physics/laureates/1914/laue-lecture.pdf Peters, P. (n.d.). W.C Roentgen and the discovery of X-rays. Retrieved November 10, 2010, from MedCyclopaedia: http://www.medcyclopaedia.com/library/radiology/chapter01.aspx Schields, P. J. (2010, January 29). Bragg's Law and Diffraction: How waves reveal the atomic structure of crystals. Retrieved November 10, 2010, from SUNY: StonyBrook: http://www.eserc.stonybrook.edu/ProjectJava/Bragg/

Endnotes

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Appendix - Raw Data Grazing Angle (2 * Deg)

Θ (Deg)

Intensity (Occurances/10s)

12

6

802

13

6.5

803

14

7

896

15

7.5

886

16

8

969

17

8.5

936

18

9

905

19

9.5

875

20

10

742

21

10.5

730

22

11

688

23

11.5

623

24

12

603

25

12.5

511

26

13

507

27

13.5

450

27.166

13.583

415

27.332

13.666

478

27.498

13.749

779

27.664

13.832

1663

27.83

13.915

2173

27.996

13.998

2335

28

14

2378 Page | 1

28.162

14.081

2049

28.328

14.164

1855

28.494

14.247

1229

28.66

14.33

958

29

14.5

678

30

15

415

30.17

15.085

437

30.336

15.168

442

30.502

15.251

501

30.668

15.334

1010

30.834

15.417

4578

31

15.5

7073

31.166

15.583

7366

31.332

15.666

6965

31.498

15.749

6055

31.664

15.832

4220

31.83

15.915

2670

32

16

1536

33

16.5

305

34

17

304

35

17.5

266

36

18

256

37

18.5

237

38

19

232

39

19.5

216

40

20

240

41

20.5

197

Page | 2

42

21

199

43

21.5

196

44

22

181

45

22.5

168

46

23

175

47

23.5

159

48

24

147

49

24.5

147

50

25

154

51

25.5

157

52

26

150

53

26.5

152

54

27

132

55

27.5

140

56

28

148

57

28.5

133

58

29

149

58.16

29.08

372

58.32

29.16

523

58.48

29.24

490

58.64

29.32

524

58.8

29.4

466

58.96

29.48

350

59

29.5

493

60

30

158

61

30.5

132

62

31

141

63

31.5

152

Page | 3

64

32

143

65

32.5

182

65.1666

32.5833

178

65.3332

32.6666

256

65.4998

32.7499

322

65.6664

32.8332

490

65.833

32.9165

954

66

33

1360

66.166

33.083

1201

66.332

33.166

773

66.498

33.249

568

66.664

33.332

334

66.83

33.415

221

67

33.5

158

68

34

108

69

34.5

145

70

35

113

71

35.5

117

72

36

126

73

36.5

106

74

37

118

75

37.5

108

76

38

137

77

38.5

106

78

39

134

79

39.5

156

80

40

116

81

40.5

137

Page | 4

82

41

118

83

41.5

132

84

42

145

85

42.5

120

86

43

151

87

43.5

123

88

44

140

89

44.5

133

90

45

115

91

45.5

122

92

46

126

93

46.5

130

94

47

138

95

47.5

229

96

48

141

97

48.5

142

98

49

115

99

49.5

113

100

50

131

101

50.5

150

102

51

170

103

51.5

156

104

52

137

105

52.5

156

106

53

127

107

53.5

138

108

54

139

108.166

54.083

146

Page | 5

dα1 2.86635E10

108.32

54.16

151

108.474

54.237

243

108.628

54.314

369

108.782

54.391

454

108.936

54.468

464

109.09

54.545

517

109.244

54.622

458

109.398

54.699

428

109.552

54.776

358

109.706

54.853

301

109.86

54.93

253

110

55

383

111

55.5

155

112

56

167

113

56.5

149

114

57

158

115

57.5

175

116

58

147

117

58.5

159

118

59

148

119

59.5

177

120

60

152

dα2 2.82756E10

dα3 2.84E-10

dβ1 2.85216E10

dβ2 2.83222E10

dβ3 2.80763E10

Avg. d 2.84E-10

** Values in the table directly above are in units of meters

Page | 6

Appendix – Disc Contents Root Directory

○ Lab 5 – Xray Diffraction Report.doc 

The official Lab Report in Microsoft Word format concerning the X-ray diffraction lab

○ Bragg Diffraction Lab.xls 

The Excel Spreadsheet used to take data points while performing the experiment. The spreadsheet also contains calculated values of d, and the graph visually depicting Intensity vs. Grazing angle

○ Fig2.gif 

Figure 2: Scanned from University Physics (12th Ed.) See 8

○ Fig3.gif 

Figure 3: Scanned from University Physics (12th Ed.) See 8

Page | 7

1

Peters, P. (n.d.). W.C Roentgen and the discovery of X-rays. Retrieved November 10, 2010, from MedCyclopaedia: http://www.medcyclopaedia.com/library/radiology/chapter01.aspx

2

Kevles, Bettyann Holtzmann (1996). Naked to the Bone Medical Imaging in the Twentieth Century. Camden, NJ: Rutgers University Press. pp. 19– 22. ISBN 0813523583.

3

"The Nobel Prize in Physics 1901". Nobelprize.org. 11 Nov 2010 http://nobelprize.org/nobel_prizes/physics/laureates/1901/

4

Laue, M. v. (1915, November 12). Concerning the Detection of X-Ray Interferences. Retrieved November 10, 2010, from NobelPrize.org: http://nobelprize.org/nobel_prizes/physics/laureates/1914/laue-lecture.pdf

5

David R. Lide, ed (1994). CRC Handbook of Chemistry and Physics 75th edition. CRC Press. pp. 10–227. ISBN 0-8493-0475-X.

6

Schields, P. J. (2010, January 29). Bragg's Law and Diffraction: How waves reveal the atomic structure of crystals. Retrieved November 10, 2010, from SUNY: StonyBrook: http://www.eserc.stonybrook.edu/ProjectJava/Bragg/

7

W. L. Bragg, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 89, No. 610 (Sep. 22, 1913), pp. 248-277

8

Hugh D. Young, R. A. (2007). University Physics. Pearson Addison-Wesley.