Physics307L:People/Klimov/Electron Diffraction Final

=On the Diffraction of Electrons Through a Graphite Foil= Author: Paul V. Klimov

Experimentalists: Paul V. Klimov, Garrett McMath

Institution: University of New Mexico

Address: University of New Mexico, Department of Physics and Astronomy, Albuquerque NM, 87106

Abstract
One century after its discovery, electron diffraction has been exploited as a useful characterization technique in many fields of physics and chemistry. In this experiment, electron diffraction was used to determine two characteristic spacings in graphite's crystal lattice. The spacings were measured to be .123(3)nm and .215(2)nm, compared to the accepted values of .123nm and .213nm, respectively. In follow up studies, the resolution and efficiency of electron diffraction can be compared to those in diffraction experiments involving other particles, such as neutrons. Such a study would facilitate future research efforts by providing a more comprehensive account of the limits of diffraction experiments involving different particles.

Introduction
The term diffraction refers to the phenomenon of wave interference. Until the early 20th century, diffraction was believed to be an effect reserved to waves, such as light. However, in 1927, an experimental 'accident' by Davidson and Germer led to the discovery that particles, too, could diffract -- that is, particles had some sort of wave nature [1]. Coincidentally, matter waves had been predicted in the doctorate thesis of Louis de Broglie, which was completed several years earlier, in 1924. de Broglie claimed that matter must behave like waves in certain limits to preserve the symmetry often observed in nature [2]. Although the first particles that were observed to diffract were electrons, it has been shown since that essentially anything can diffract, given the correct conditions. This was also something predicted in de Broglie's thesis. In 1929, de Broglie received the Nobel Prize in Physics, for his work on matter waves and wave mechanics in general.

Over the course of the century, particle diffraction has become a huge component of various fields of physics and chemistry. The applications of electron diffraction, and matter diffraction in general, are absolutely boundless. One might inquire as to why matter diffraction is so much more useful that light diffraction, and the answer is crystal clear. Not only are electrons easily produced, but they are easily accelerated to short wavelengths that are unrivaled by photons. This well known fact has found its way into the fundamental workings of transmission electron microscopy, biological electron microscopy [3], short wavelength crystallography, and many other applications. Given the practical and historical significance of electron diffraction, it is a useful experiment to repeat not only to test the theory, but to embrace the beauty of its result.

In this experiment, electrons were accelerated through an atomic diffraction grating - graphite foil. The diffracted electrons were projected onto a fluorescent screen where the diffraction maxima could easily be seen. Given that graphite is highly polycrystalline in nature, the diffraction pattern formed as a ring, and not dots as would be expected from a single-crystal. The diffraction rings were measured and their diameters were then related to the atomic spacings (see Figure 2 for diagram). Given the geometry of our experimental device, only the first order diffraction maxima were visible, and thus measured.

Materials and Methods




The electron diffractor (model 2555, 5kV, .3mA ratings) consists of several key components. The first is the high resistance filament, which is responsible for ejecting electrons. The filament was powered a Teltron Limited power supply (London England 813KV), which was wired in series with a Wavetek-Materman 85XT ammeter, to monitor the input current. Just past the filament, a voltage was set up to accelerate the ejected electrons. This potential was also produced by the Teltron power supply. Immediately after the anode resides the graphite foil, which serves as the atomic diffraction grating and as an entrance for the electrons into the spherical vacuum bulb. The bulb is coated in a fluorescent compound, which was activated upon collisions with electrons. A schematic of the diffraction device is provided to the right in Figure 1.

The electron diffraction device used in this set of experiments was extremely fragile. And in particular, the graphite foil used as the diffraction target was incredibly thin and it was given that excessive electron flux / current could easily puncture this foil. This was acknowledged and current was monitored at all time, not to let it exceed .25mA, which was just below the current rating of the diffraction device.

Several minutes after turning on the device, the diffraction patterns become visible on the fluorescent screen. Due to the fact that graphite is highly pollycrystaline, the diffraction patterns are rings and not dots, as would be expected otherwise. Given that the diffraction rings had finite width, we decided to measure only the inner diameters of each ring to avoid ambiguity and added error. The diameter of each ring was measured in this way using Carrera Percision digital calipers, accurate to one hundredth of a millimeter. This measuring technique was administered as we varied the accelerating potential from 5kV until roughly 3.3kV, at which point the diffraction rings had faded significantly, and we decided that accurate measurements could no longer be obtained.

After taking the measurements by the method described above, the atomic lattice spacings were calculated from the following relationship (see Appendix for an in-depth derivation and the list of used constants):


 * $$ d_{j}=\frac{2h (L_{o} + R_{c}(1-Cos(ArcSin(\frac{D_{j}}{2R_{c}})))}{D_{j}\sqrt{2meV_{a}}} - t \frac{D_{j}}{L_{o}} $$

$$ R_{c}$$ is the radius of curvature of the bulb, $$D_{j} $$ is the diameter of the diffraction ring in question, $$V_{a}$$ is the accelerating voltage, $$L_{o}$$ is the distance from the graphite to the opposite end of the vacuum bulb, t is the thickness of the bulb, h is the Planck constant, m is the electron mass, and e is the electron charge.

The equation is fairly complicated due to the fact that several corrections were made to account for the geometry of the tube. The fact that the graphite foil is not at the center of the vacuum bulb and that the bulb has finite thickness introduced the need for corrections.

Data analysis was performed using MATLAB's built in algorithms, unless otherwise specified. 

Tables and Figures
Table 1: Measurements of the outer and inner diffraction maxima, for each respective accelerating voltage V. Uncertainties were prescribed for each measurement based on the approximate width of the diffraction rings. The inner diameter of the outer maxima and the inner diameter of the inner maxima were measured. Measurements cease at 3200V where the diffraction rings are nearly impossible to see. The calculated atomic spacings from each datapoint are given in the last two columns, for each respective diffraction ring. The given uncertainties are the propagated error for each respective data point, given by: $$ \Delta d = d \Delta D/D $$. We believe that this uncertainty is the best 95% confidence interval.

Results and Discussion


Using the methods developed and the measurements given above, the atomic spacings were calculated for each datum, and are given in Table 1. Uncertainties corresponding to the diameter of the diffraction maxima were propagated to give the SEM. These data are also plotted in Figures 3 through 6, with corresponding least squares fits, means, and standard errors of the means where appropriate. Using the means, we see that the accepted values of the atomic spacings both lie within the given 68% confidence interval.


 * $$ d_{1}^{exp}=1.23(3) \AA $$


 * $$ d_{2}^{exp}=2.15(2) \AA $$


 * $$ d_{1}^{act}=1.23 \AA $$


 * $$ d_{2}^{act}=2.13 \AA $$

Interestingly, measurements of the outer diameter seem to follow a non-linear trend (see Figure 3 and 5), which is likely the result of some systematic error. A possible, though fairly unlikely explanation for this error is a power supply malfunction, which would have caused us to undershoot some voltage measurements. Although this is an unresolved issue because the accelerating potential was not measured on a secondary voltmeter, it is our contention that the power supply was working properly. The reason being that to account for the trend that we saw, one would have to undershoot voltage measurements on the order of 1kV, which is highly unlikely.

The more reasonable source of systematic error involves the apparent widening of the diffraction rings. As the rings expand due to lower accelerating voltages, the angle of incidence between the bulb and the electron beam grows. Because the electrons are not accelerated from the center of the spherical bulb, this causes the diffraction rings to appear wider then they actually are (see Figure 7). Because our diffraction rings were measured at ~2mm wide, this apparent widening could reasonably affect measurements by roughly 1mm, which is double the error for each radius. The geometry involved suggests that the errors should progress non-linearly, which could very well account for the s-shaped curve that was obtained. This claim is further supported by our measurements of the inner diffraction ring, for which a non-linear trend was not seen (see Figures 4 and 6). The inner ring is always small in diameter; therefore, its angle of incidence with the vacuum bulb is always approximately perpendicular. This would cause the width of the diffraction ring to stay essentially constant for the various accelerating voltages that were used.

Despite the possibility of systematic errors involving the widths of the diffraction rings, our data returned values of graphite's atomic spacings which are consistent with the literature. The inner spacing was measured at .123(3)nm and the outer at .215(2)nm, as compared to the accepted .123nm and .213nm, respectively. We should expect consistent results involving the smaller diffraction ring (corresponding to the larger spacing) since measurements involving this ring were less susceptible to errors involving the apparent widening of the rings. The outer diffraction ring, however, should have returned inconsistent results due to systematic error. For this reason, it is possible that our outcome was the result of some luck. In Figure 5 we see that our first measurements are consistently above the corresponding least squares line. As the accelerating voltage dropped and the rings expanded, our measurements of the diameter came with smaller changes. Eventually, this caused our data to fall below the trendline. Presumably these later data counteracted our initial data, providing us with an average that is close to the accepted value.

To avoid systematic errors related to the widening of the diffraction rings in the future, one could set up a diffraction apparatus with the target foil at the center of the spherical bulb. This would cause electrons to always strike the glass envelope at a 90 degree angle.

Conclusions
Our results were consistent with theoretical predictions and the experimentally accepted values of the characteristic atomic spacings in graphite. This result shows the power of the electron diffraction technique in the field of crystallography. Even with limited and outdated equipment, we were able to obtain a great measurement of the spacings in graphite. It is hard to imagine the kind of accuracy possible with modern equipment.

From a theoretical perspective, electron diffraction clearly portrays the accuracy of de Broglie's wave theory. Experimental confirmation of this theory is crucial in physics because the de Broglie wave theory is really the basis of quantum/wave mechanics. In this experiment, entities which have classically been considered particles, electrons, clearly behaved like waves.

While work presented here shows only one of the many applications of electron diffraction, it is perhaps the most fundamental. The same basic principles that were used here apply to all other analysis techniques involving particle diffraction. In follow up experiments, one might try to compare the resolving power of electron diffraction to other forms of diffraction, possibly including other particles or high energy photons. Such work would have direct relevance to crystallography.

Acknowledgements
I would like to thank my partner, Garrett McMath, our helpful TA, Aram Gragossian, and of course Dr.Koch who provided us with ideas and helped us out throughout the various experiments. I would also like to thank the University of New Mexico for allowing us to use one of their labs to perform this experiment. 

MATLAB CODE
All results and data were generated in MATLAB. The code used is given here: [[Media:ElectDiff_PKlimov.m | Electron Diffraction MATLAB Code]]

Derivations of used formulas
The kinetic energy gained by the electrons as they are accelerated through a potential difference of Va is given by the work-kinetic energy theorem:

$$ \Delta E_{kin}=-\Delta U = -\int_{c} \mathbf{F} \cdot d\mathbf{r} = \int_{c} e\mathbf{E} \cdot d\mathbf{r} = e(\mathbf{E\cdot \hat{r}}) \int_{c} dr = eE(r_{2}-r_{1})=eV_{a} $$

Where i have made the approximation that fringing of the electric field is negligible, which is a reasonable approximation, given that the spacing between the anode and cathode is fairly small. Then, making use of the de Broglie theory of matter waves, we can now say that the accelerated electrons have a wavelength of:

$$ \lambda_{dB} = \frac{h}{|\mathbf{p}|}=\frac{h}{p}=\frac{h}{\sqrt{2mE_{kin}}}=\frac{h}{\sqrt{2meVa}} $$

Given that electron energies were generally on the order of around 4keV, there will be no benefit in the computation of a relativistic deBroglie wavelength.

At this point, we must invoke the Bragg theory of diffraction, which will allow us to relate the atomic spacings in the crystalline target to the produced diffraction pattern. The Bragg condition is given by the following relationship. 'd' is the lattice spacing, n is the diffraction order, theta is the angle between two atoms in adjacent rows of a lattice, and lambda is the wavelength of the wave.

$$ 2\cdot d\cdot sin(\theta_{max})=n \cdot \lambda_{dB} $$

This is general statement about the 'reflection' of matter waves from atoms in the targets lattice. The equation says that the path difference between two matter waves must be equal to an integer number of wavelengths. This condition forces the outgoing waves to interfere in-phase at the angle $$ \theta_{max} $$. Likewise, one could find the diffraction minima by setting the phase difference equal to an odd number of wavelengths, forcing the interference to be destructive.

Given the geometry of our device, we see that the angle subtended from the crystal to the diffraction maxima turns out to be $$ 2\theta $$, conveniently relating what we see on the experimental level to the atomic scale. We now must use some small angle approximations to arrive at an expression that can be used in determining the atomic spacings. R is the radius of the diffraction maxima and L is the distance from the foil to the screen.

$$ \frac{R}{L}= sin(2\theta) \approx 2sin(\theta) $$

Now, setting the diffraction order to 1 (n=1), we can relate these quantities to the deBrogle wavelength:

$$ d (2sin(\theta))=(1)\lambda_{dB}=\frac{Rd}{L} $$

Combining the above relationships, and using the fact that R = D/2, the diameter of the ring, we arrive at the important equation relating measurable quantities to the atomic spacings:

$$ \frac{Rd}{L}=\frac{h}{\sqrt{2meV_{a}}} \Rightarrow d=\frac{2h L}{D\sqrt{2meV_{a}}} $$

However, this equation is not complete given that the apparent length of the device, L, will be a function of the angle of diffraction. This quantity is easily related to the diameter of the diffraction pattern :

$$ L = L(D) $$

$$ d=\frac{2h L(D)}{D\sqrt{2meV_{a}}} $$

The relationship was found, which also invokes the radius of curvature, $$ R_{c} $$ of the spherical glass envelope. The quantity $$ L_{o} $$ is the distance from the graphite foil to the center of the glass envelope:

$$ L(D) = L_{o} + R_{c}(1-Cos(ArcSin(\frac{D}{2R_{c}})) $$

Yet another correction will have to be made for the thickness of the glass envelope, t, which is given at 1.5mm. If this is not accounted for, our measurement of the diffraction ring diameter will too large by the following factor:

$$ 2 t sin(\theta) = 2 t \frac{D}{2L} $$

Therefore, the final relationship relating atomic spacings and the diameter of the diffraction maxima, including all corrections is:

$$ d=\frac{2h (L_{o} + R_{c}(1-Cos(ArcSin(\frac{D}{2R_{c}})))}{D\sqrt{2meV_{a}}} - t \frac{D}{L} $$

This is the equation used in the calculation of all atomic spacings.

List of Used Constants
$$ h = 6.626 \cdot 10^{-34} J \cdot s $$

$$ e = 1.6 \cdot 10^{-19} C $$

$$ m = 9.11 \cdot 10^{-31} kg$$

$$ R_{c}= 66 \cdot mm$$

$$ L_{o}=.130 \cdot m$$

$$ t= 1.5 \cdot mm$$