X- Ray Powder Diffraction.

Using this technique, we can obtain our coherent monochromatic light source. We are given that the sodium chloride (NaCl) and the potassium chloride (KCl) have a cubic crystalline system, given by

sin2? = ?2 (h2 + k2+ l2) (2)

4a2

where "a" is the unit cell dimension, the constant that we seek in the experiment and has a geometric representation given in Figure 2 and h, k and l are the Miller Indices of the plane. The miller index is the relative spacing of a unit cell with respect to another unit cell.

Fig.2 lattice structure of cubic crystals.

These cubic lattices are three equal axes at right angles. Other crystalline solids have structures such as Tetragonal (three axes at right angles, two equal), Orthorhombic (three unequal axes at right angles), Rhombohedral (three equal axes, equally inclined), Hexagonal (two equal coplanar axes at 120°, third axis at right angles), Monoclinic (three unequal axes, one pair not at right angles) and Triclinic which has three unequal axes, unequally inclined and none at right angles [1].

The Debye-Scherrer camera, seen in Figure 3, has a diameter of 57.3 mm, which allows for easy calculations, since 57.3 ~ 180/ ? (inverse radian).

Figure 3: the Debye-Scherrer camera and its components.

The camera is loaded with Kodak Direct Exposure film. This film is coated with a special gelatin silver halide emulsion on both sides of a very fine grain size- the finer the grain, the greater the resolution. When developing the film, the developer affects the particles that have interacted with the x rays. These grains then undergo a chemical reaction that is stopped when placed in the "stop-bath". Without this bath, all the grains would turn dark and the film would give us no information. After the stop-bath, the film is placed in a fixer to react and change colour. The film is then placed in a water bath to rinse off the chemicals. The lines are then visible and readily measured once the film is dry.

To obtain our lattice constant "a" we square equation (1) and solve for sin2?.

n2 ?2 = 4d2 sin2(?), (n =1) (1')

sin2(?) = ?2

4d2

sin2? = ?2 (h2 + k2+ l2) (2)

4a2

:. ?2 = ?2 (h2 + k2+ l2)

4d2 4a2

and we can see that this gives

a2 = d2 (h2 + k2+ l2) (3)

We can solve (1) for d and find that d = ?/2sin?. Therefore, (3) becomes

a = ? (h2 + k2+ l2)1/2 (4)

2sin?

When the film in developed, two distinct sets of concentric circles will be visible. The diameter of the camera lets us easily read that 2mm is 1°.

Figure 4: diffraction patterns on the film

Experimental Procedure

Our procedure consists of 6 trials: 1-3 with NaCl, 4-6 with KCl. The following steps were used to obtain our results.

Trial 1(General): We crushed the NaCl into a medium-coarse powder. We took a thick hollow glass rod an applied the flame from a torch to stretch the glass to a very thin glass rod. We then applied a very small amount of vacuum grease to a Kim Wipe and rolled a glass fragment (~2-3mm long) in this grease, ensuring an even coat. Excess grease was removed, and the sample was emerged in the freshly ground powder. Fresh powder is used in every trial to ensure humidity and impurities in the air don't affect our results significantly. A piece of putty was placed in the center of the Debye-Scherrer camera and the sample was mounted inside. The sample was then aligned with the help of a small telescope piece. The sample was manually rotated and adjusted until the image through the telescope did not move up or down when turned. Once the alignment was complete, the Kodak Direct Exposure Film was brought into a completely dark room with the camera and cutting device. In total darkness, the box was opened and the film was cut to ~57.3mm with the help of pre-existing notches in the cutter. The film was loaded into the camera and the camera was shut. A piece of fabric was placed on the camera as to minimize light entering the camera through the x ray portal. The camera was then placed on the x ray machine and tightened. The exposure time was set to 20 minutes at 30kV and 23mA.

During the exposure, the developing process was set up. It consisted of four baths: developer, stopper, fixer and rinsing water. It is important to not let the chemicals sit open for too long, as they react with oxygen as well.

After the exposure was complete, the camera was removed and brought back into the dark room where the developing process is performed. Dip the film in the developer for 30s while gently agitating. Then move the film into the stop bath for another 30s. Place the film in the fixer for 2mins, still agitating and then place it in a water bath. After the film has dried, the rings should be visible and readily measured with a vernier caliper. Place the film on a light table to make the lines more visible.

Trial 2: repeat trial 1 with a fine NaCl powder with a 30min exposure.

Trial 3: repeat trial 1 with semi-fine NaCl powder and expose for 1 hour.

Trial 4: repeat trial 1 with KCl in a semi-fine powder for 20 min exposure.

Trial 5: repeat trial 4 with 1 hour exposure with fresh fixer and developer

Trial 6: repeat trial 4 with coarser powder and 45 min exposure.

With all the films, measure the diameter of the rings and using equations 1, 2 and 4 we can calculate the experimental value for d. Then, using table 7, compare it to our theoretical values, as seen in Table 1 (Appendix 1). In order to determine the miller indices (h k l) Table 5-3 [3] was consulted once experimental "d" values were found. Using these educated guesses and the knowledge that cubic structures can only take on values of (1 1 1), (2 0 0), (2 2 0), (2 2 2) or (4 0 0) we were able to determine the possible plane and solve for "a".

Experimental Results:

All trials showed strong reflected lines and poor transmitted lines.

Trial 1 with the NaCl powder showed strong reflected lines and poor transmitted ones. The two darkest rings were rings # 2 and #4 (on the reflected side) correspond to confirmed planes with miller indices of (2 2 0) and (2 0 0) respectively, shown in Table 1 (appendix 1).

Trial 2 showed more planes than the first trial as seen in Table 2, however, these planes ended up being meaningless as the only two planes that agreed with known values were rings #3 and #6 (on the reflected side) which are the same planes seen in trial 1 of

(2 2 0) and (2 0 0) respectively. The transmitted side showed more lines than the previous film, but none of them were useful.

The 3rd trial gave the same (2 2 0) and (2 0 0) planes from the reflected side of the film as before but added a new (1 1 1) plane to our results on the transmitted side seen in Table 3. One can also observe small dots in the film appearing between the rings. This will be discussed later.

Trial 4 with the KCl gave an inconclusive piece of film. One can see (very carefully) small spots and pieces of rings, but none that can be measured. Trial 4 did not contribute to the results.

The next trial with KCl, trial 5, gave very clear and distinct planes. Rings 4 and 7 appeared to be the most well defined, however, their distance values (seen in Table 4) matched them up with miller planes of (2 2 1) (or (3 0 0) you can't tell since they both sum squared to 9) and (2 1 0). These planes are not possible and must therefore be discarded.

Trial 6 gave very patchy circles, the cause of which will be explored shortly. It also appears that there may have been a misalignment when loading the film into the camera, as the holes in the film do not exactly line up with the origin of the rings. These results are unusable.

Hence, from our (2 2 0), (2 0 0) and (1 1 1) results from NaCl, a value for a was calculated to be a = 6.169 ± 2.205 Å. Our given value for a (NaCl) was 5.369 Å. Taking the difference between our given and experimental values, and dividing by the deviation,

?? = (6.169-5.369)/2.205

= 0.24 <2

Therefore we can conclude that our result is statistically significant.

Discussion:

As seen in Tables 1-3, the planes with miller indices (2 2 0) and (2 0 0) are repeated in all trials for NaCl, and (1 1 1) just barely makes it as seen in trial 3. None of the "observed" planes for KCl were actual planes according to the cubic lattice rule. That is, cubic structures can only take on values of (1 1 1), (2 0 0), (2 2 0), (2 2 2) or (4 0 0) due to symmetries.

Trials 6 and 3 have a strange "dot" phenomenon on the film. This may be due to coarse powder: instead of there being many small fragments of substance, there is only one large chunk. This large piece can only align with one plane, whereas the other smaller pieces that it could have made would have lined up nicely with several planes when rotated. This is why it causes a discontinuous line instead of a smooth one.

It is possible that in trial 4, the powder was so coarse that no diffraction lines could be recorded. If one studies the film closely you will see very faint discontinuous lines forming light circular patterns. There also could have been a problem with our chemicals when developing the film, since they are reactive with oxygen. When the developer was replaced and new fixer was mixed, the following results turned out well. Therefore, it could have been a combination of overly coarse powder and overused chemicals that lead to the failure of trial #4 with KCl.

It should also be noted that ideally, this experiment should be conducted in a temperature-controlled room. As a result of varying temperatures on surfaces the film can shrink. More specifically, the film has certain temperature inside its casing, but is then exposed to an outer temperature on the cutting board, a different temperature inside the camera, different temperatures in the developing solutions and another temperature change when it is laid out to be measured once it has dried. Drying also causes shrinkage.

Another source of error in our experiment can be from the divergence of our beam. Should the beam be ever so slightly divergent, it would cause an uneven reduction of intensity of the reflected beams [1].

X rays can also be attenuated by the film itself once it passes through the reflected side and is transmitted. This explains why the transmitted side is always more faint than the reflected side.

The other rings that don't fit our calculations of where planes should be could be caused by impurities in the sample. The different thicknesses of the rings could be caused by interference from Kß rays that got through the nickel attenuator.

Once our data has been fully analyzed, we see that from Table 6 our value for NaCl "a" = 6.169 ± 2.205 Å. Our given value for "a" (NaCl) was 5.369 Å. This falls well within our acceptable experimental range of results (± 2?) at 0.24? with full error calculations seen in Appendix 2. The experiment would need to be repeated to obtain more values for KCl.

References:

[1] Laing, Michael: "An Introduction to the Scope, Potential and Applications of X-ray Analysis", http://www.iucr.ac.uk/iucr-top/comm/cteach/pamphlets/2/node1.html

October 13, 2003.

[2]Winter, W.T. "Polymer Techniques Lab Expt. F: X-ray Diffraction", http://www.esf.edu/chemistry/winter/FCH551/xraylab.pdf, October 13, 2003.

[3] Nuffield, "X-Ray diffraction methods", John Wiley and Sons Inc., NY, 1966.

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