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

Figure 1: The first X-Ray Scanhttp://en.wikipedia.org/wiki/File:Anna_Berthe_Roentgen.gif

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 him the first Nobel Prize in Physics ever bestowed3

(1901), and paved the way for countless applications 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

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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 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

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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 X-ray generation, it is

only fitting to briefly describe how this is acquired. In very simple terms, 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.

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

X-ray. 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

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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

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 phase-difference 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

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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 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-to-trough), 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,

sin (θ )= oppositehypotenuse

= ABd→ AB=d∗sin (θ ) (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:

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nλ=2 AB (eqn. 3)

All that is left is to substitute equation 2 into equation 3 and we achieve:

2d∗sin (θ ) = n λ (eqn. 4 - Bragg’s Law)

N = An integer number (1, 2, 3, 4, …)d = The distance from atom-to-atom inside the moleculeλ = The wavelength of the X-Ray used during the scatteringθ = 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 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

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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 user 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 devices on the stage and shielding the user 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

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diameter slots (1mm – 3mm) for narrowing the beam. The primary collimator makes use of 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 X-ray occurrences over a user-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 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

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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 Tel-X-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 user-

friendly, and only required the push of a button at each angle to perform a ten second

measurement of the 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

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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 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.

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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:

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:

Graz ing 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:

GraphData=Sheet1 !$ B$ 4 : $C $159

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0 10 20 30 40 50 60 700

1000

2000

3000

4000

5000

6000

7000

8000Intensity vs. Angle

Angle (Degrees)

Inte

nsit

y (O

ccur

ance

s/10

sec)

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 account and apply Bragg’s law to it as follows:

1∗(1.54 x10−10m )=2∗d∗sin (15.583° )→d=(1.54 x 10−10m )2∗sin (15.583 ° )

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

dα 1=2.866 x 10−10m≈0.286nm

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.

d β1=(1.38 x 10−10m)2∗sin (13.998 ° )

=2.853 x10−10=0.285nm

In the following discussion section, we will take a careful look at all of our d-values 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=∑0

n

dn

n

(3)

In (3), n represents the total number of data points. Continuing with our sample calculations,

we will use the determined values of dα 1and d β1 and find their average as follows:

d1=0.285nm+0.286nm

2=0.2855nm

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)2

n

(4)

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This is important in observing how our data varies, and we achieve a sample deviation using the

two previous values and their average below:

σ=√ (0.286nm−0.2855nm)2+(0.285nm−0.2855nm)2

2=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:

|.|Error=¿dreal−dexperiment | (5)

Therefore, using (5), the absolute error of our first alpha peak is as follows:

|.282nm−.286nm|=.004 nm

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=|.|Errordreal

(6)

%Rel .Error= (Rel . Error )∗100% (7)

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

Rel . Error=0.004nm0.282nm

=0.0142

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%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. (Remember 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).

dα 1=(1 )∗(0.154 x10−9m )2∗sin (15.583 ° )

=2.86635 x 10−10m=0.2867nm

d β1=(1 )∗(0.138x 10−9m )2∗sin (13.998 ° )

=2.85256 x10−10m=0.2853nm

dα 2=(2 )∗(0.154 x10−9m)

2∗sin (33 ° )=2.82756 x 10−10m=0.2828nm

d β2=(2 )∗(0.138x 10−9m )2∗sin (29.32 ° )

=2.81813 x10−10m=0.2818nm

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dα 3=(3 )∗(0.154 x10−9m )2∗sin(54.391 °)

=2.8413x 10−10m=0.2841nm

d β3=(3 )∗(0.138 x10−9m )2∗sin (47.5 °)

=2.80763 x10−10m=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=√ (0.2867−0.282 )2+(0.2853−0.282 )2+(0.2828−0.282 )2+(0.2818−0.282 )2+ (0.2841−0.282 )2+(0.2808−0.282 )2

6∗10−9

¿2.566x10-12

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):

|.|Error=|0.282nm−0.2836 nm|=0.00158nm

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

%Rel .Error=0.00158nm0.282nm

∗100%=0.5615% error

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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.

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.

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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

atomic-separation indirectly by use of electromagnetic radiation. This is due to the 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

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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/

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Endnotes

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Appendix - Raw Data

Grazing Angle (2 * Deg) Θ (Deg) Intensity (Occurances/10s)

12 6 80213 6.5 80314 7 89615 7.5 88616 8 96917 8.5 93618 9 90519 9.5 87520 10 74221 10.5 73022 11 68823 11.5 62324 12 60325 12.5 51126 13 50727 13.5 450

27.166 13.583 41527.332 13.666 47827.498 13.749 77927.664 13.832 166327.83 13.915 2173

27.996 13.998 233528 14 2378

28.162 14.081 204928.328 14.164 185528.494 14.247 122928.66 14.33 958

29 14.5 67830 15 415

30.17 15.085 43730.336 15.168 44230.502 15.251 50130.668 15.334 101030.834 15.417 4578

31 15.5 707331.166 15.583 736631.332 15.666 696531.498 15.749 605531.664 15.832 422031.83 15.915 2670

32 16 153633 16.5 305

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34 17 30435 17.5 26636 18 25637 18.5 23738 19 23239 19.5 21640 20 24041 20.5 19742 21 19943 21.5 19644 22 18145 22.5 16846 23 17547 23.5 15948 24 14749 24.5 14750 25 15451 25.5 15752 26 15053 26.5 15254 27 13255 27.5 14056 28 14857 28.5 13358 29 149

58.16 29.08 37258.32 29.16 52358.48 29.24 49058.64 29.32 52458.8 29.4 466

58.96 29.48 35059 29.5 49360 30 15861 30.5 13262 31 14163 31.5 15264 32 14365 32.5 182

65.1666 32.5833 17865.3332 32.6666 25665.4998 32.7499 32265.6664 32.8332 49065.833 32.9165 954

66 33 136066.166 33.083 120166.332 33.166 77366.498 33.249 56866.664 33.332 334

Page | 23

66.83 33.415 22167 33.5 15868 34 10869 34.5 14570 35 11371 35.5 11772 36 12673 36.5 10674 37 11875 37.5 10876 38 13777 38.5 10678 39 13479 39.5 15680 40 11681 40.5 13782 41 11883 41.5 13284 42 14585 42.5 12086 43 15187 43.5 12388 44 14089 44.5 13390 45 11591 45.5 12292 46 12693 46.5 13094 47 13895 47.5 22996 48 14197 48.5 14298 49 11599 49.5 113

100 50 131101 50.5 150102 51 170103 51.5 156104 52 137105 52.5 156106 53 127107 53.5 138108 54 139

108.166 54.083 146108.32 54.16 151

108.474 54.237 243108.628 54.314 369108.782 54.391 454

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108.936 54.468 464109.09 54.545 517

109.244 54.622 458109.398 54.699 428109.552 54.776 358109.706 54.853 301109.86 54.93 253

110 55 383111 55.5 155112 56 167113 56.5 149114 57 158115 57.5 175116 58 147117 58.5 159118 59 148119 59.5 177120 60 152

dα1 dα2 dα3 dβ1 dβ2 dβ3 Avg. d2.86635E-10 2.82756E-10 2.84E-10 2.85216E-10 2.83222E-10 2.80763E-10 2.84E-10

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

Page | 25

Appendix – Disc Contents

Root Directory

o Lab 5 – Xray Diffraction Report.doc

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

o 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

o Fig2.gif

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

o Fig3.gif

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

Page | 26

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.