• N. W. Ashcroft and N. D. Mermin. _Solid State Physics, _W. B. Saunders Co. (1976).
• C. Kittel. _Introduction to Solid State Physics, _John Wiley & Sons, Inc. (1996).

In this experiment you will test the de Broglie relation by analyzing the diffraction pattern formed when electrons of known momentum are scattered from graphite crystals. Your goal is to demonstrate that electrons have a wave-like nature as predicted by de Broglie.

Recall that when an electron is accelerated from rest, through a potential difference V, its kinetic energy is increased by eV so that

{ $eV = p^2/2m$ { $eV = p^2/2m$

(1)

where { $e$ { $e$  is the charge of the electron, and where { $p$ { $p$  and { $m$ { $m$  are the momentum and rest mass of the electron. Eq. (1) is true for non-relativistic electrons. Is this a good assumption in your experiment? In 1924 L. de Broglie proposed that particles should exhibit wave-like behavior, the particle wavelength being inversely proportional to its momentum according to the relation,

{ $\lambda = h/p$ { $\lambda = h/p$

(2)

where { $\lambda$ { $\lambda$ is the particle wavelength, { $p$ { $p$  is the particle momentum, and { $h$ { $h$  is Planck's constant. In 1927, Davisson and Germer scattered electrons from a nickel crystal and observed that the scattered electrons obeyed the Bragg law for diffraction from a crystal lattice with wavelengths as predicted by de Broglie's relation. Combining Eqs. (1) and (2) gives

{ $\lambda = \dfrac{h}{\sqrt{2meV}}$ { $\lambda = \dfrac{h}{\sqrt{2meV}}$

(3)

Waves “reflecting” from parallel planes (as in x-ray diffraction from crystals) may be represented as in Fig. 1.

{ ${/download/attachments/164078099/Bragg%20Planes.png?version=1&modificationDate=1565989518000&api=v2}$ Figure 1: Bragg reflection of waves from parallel reflecting planes. The extra distance traveled by light reflecting off of a successive plane ({ $2d\sin\theta$ { $2d\sin\theta$ ) is indicated in green. In a crystal lattice the reflecting planes are not continuous as drawn, but are actually planes defined by relatively large concentrations of atoms. The waves are elastically scattered from the atoms in the lattice. Proper calculation of phase and amplitude shows that intensity maxima occur only when the angles θ of incidence and reflection are equal, and when

{ $2d\sin\theta = n\lambda$ { $2d\sin\theta = n\lambda$

(4)

where { $\lambda$ { $\lambda$  is the wavelength of the waves reflecting from the planes, { $d$ { $d$  is the distance between the reflecting planes and { $n$ { $n$  is an integer, { $n = 0,1,2,\dots$ { $n = 0,1,2,\dots$ If electrons do behave as waves, and if the de Broglie model is correct, then the relation between the diffraction angle and the accelerating potential may be obtained by combining Eqs. (3) and (4), giving

{ $2d\sin\theta = n\dfrac{h}{\sqrt{2meV}}$ { $2d\sin\theta = n\dfrac{h}{\sqrt{2meV}}$

(5)

We will test Eq. (5) and its underlying assumptions by attempting to observe diffraction and by measuring the diffraction angle as a function of accelerating potential. If Eq. (5) stands the test, then the de Broglie relation, { $\lambda = h/p$ { $\lambda = h/p$  also stands the test.

Pyrolytic graphite, used in our apparatus, is composed of flat, parallel sheets of carbon atoms. Each sheet consists of a close packed hexagonal lattice as shown in Fig. 2.

{ ${/download/attachments/164078099/Graphene%20Spacing.png?version=1&modificationDate=1565972448000&api=v2}$ Figure 2: Hexagonal close-packed structure of carbon atoms within
one graphite sheet ({ $a = 2.4612 \times 10^{-10} \mathrm{\;m}$ { $a = 2.4612 \times 10^{-10} \mathrm{\;m}$ ). The spacing of the carbon atoms within the same sheet, shown in Fig. 2, is { $a = 2.4612 \times 10^{-10} \mathrm{\;m}$ { $a = 2.4612 \times 10^{-10} \mathrm{\;m}$ . Three-dimensional crystals are formed by stacking multiple sheets as shown in Fig. 3. { ${/download/attachments/164078099/Graphene%20Layers.png?version=3&modificationDate=1565981760000&api=v2}$ Figure 3: 3-D graphite crystal showing planes of carbon atoms.  Note that the middle layer is offset compared to the other two

In our apparatus, the graphite is in the form of randomly oriented micro-crystals. Only those crystals which happen to be oriented so as to obey the Bragg condition will participate in giving diffraction maxima.

Figure 3 shows that adjacent planes are offset from each other. Thus, the crystal pattern repeats with a periodicity of two planes. Careful examination of the first and third carbon sheets reveals rows of atoms (colored black in Fig. 3) that form equally spaced, horizontal planes, perpendicular to the carbon sheets of Fig. 2. Figure 4 shows an edge-on view of the equally spaced planes, which have the largest spacings, and therefore yield the smallest diffraction angles.

The planes shown in Fig. 4 contain large numbers of atoms and produce strong constructive interference. Cases (a) and (b) show the planes that produce the two innermost diffraction rings from graphite.

{ ${/download/attachments/164078099/Graphene%20Planes.png?version=1&modificationDate=1565981035000&api=v2}$\\

Figure 4: Pyrolitic graphite, viewed perpendicular to the carbon sheet planes. The two sets of periodic planes having
the largest spacings are shown in green and purple, where { $d_1 = 2.13 \times 10^{-10} \mathrm{\;m}$ { $d_1 = 2.13 \times 10^{-10} \mathrm{\;m}$  and { $d_2 = 1.23 \times 10^{-10} \mathrm{\;m}$ { $d_2 = 1.23 \times 10^{-10} \mathrm{\;m}$ When the electron beam is incident on the graphite crystals, electrons will be diffracted into specific angles and bright spots will appear on the phosphor screen where the diffracted electrons strike. (See Fig. 5). If the electron beam diffracts from a single crystal, the resulting diffraction pattern will be as shown in Fig 5(a).  The bright spot in the center of the pattern corresponds to _n _= 0 (zeroth order diffraction; i.e. electrons passing through without scattering). The hexagonal patterns of diffraction spots at the two smallest radii are due to the spacings { $d_1$ { $d_1$  and { $d_2$ { $d_2$  respectively, shown in Fig. 4. Larger radii result from higher order diffraction of these spacings. { ${/download/attachments/164078099/Fig_5a.png?version=1&modificationDate=1499785233000&api=v2}${ ${/download/attachments/164078099/Fig_5b.png?version=1&modificationDate=1499785237000&api=v2}$ Figure 5: Electron diffraction patterns from graphite. Figure (a) is the result of diffraction from a single crystal. Figure (b) is produced when the electron beam passes through multiple crystals, with crystal axes all oriented parallel to the tube axis, but with random orientations about that axis. Note that the axis of the electron beam is normal to the plane of the page.

The apparatus is shown in Fig. 6.  It consists of a cathode ray tube, containing a thermionic emission electron gun. The gun creates an electron beam, which is accelerated and focused through an aperture in the anode.

{ ${/download/attachments/164078099/Electron-Diffraction-Tube.png?version=1&modificationDate=1569955361000&api=v2}$\\

Figure 6: Electron diffraction tube.

The geometric details of the tube are shown in Fig. 7.

{ ${/download/attachments/164078099/Electron%20diffraction%20tube%20geometry.png?version=4&modificationDate=1570033972000&api=v2}$\\

Figure 7: Electron diffraction tube geometry.

Note from Fig. 1 that the angle between the electron beam and the diffraction spot is { $2\theta$ { $2\theta$ . To a very good approximation { $\phi \approx 4\theta$ { $\phi \approx 4\theta$ .

With the power supply turned off, make all the electrical connections as shown in Fig. 8. The function of each of these connections is shown in Fig. 6.

CAUTION!
Shock Hazard!

High voltages up to 5 kV are present when the apparatus is turned on. Make all electrical connections with the power turned off. Do not remove any wires while the power supply is turned on.

{ ${/download/attachments/164078099/Electrical%20connections.png?version=2&modificationDate=1570034122000&api=v2}$ Figure 8: Electrical connections, with the color coding the same as in Fig. 6.  Note that “Ground” is connected to the metallic exterior of the power supply and any exposed parts of the tube to prevent risk of shock; it is not the same as 0V.

Since applying a high voltage to a cold filament may cause damage to the filament, allow the filament to warm up first, as follows:

• Before turning on the power supply, reduce the accelerating potential to its minimum setting.
• Turn on the power switch and wait one minute for filament warm up.

In order to prolong the life of the phosphor screen and the graphite target, turn the tube on only while taking data. Monitor the graphite target for any redness (over-heating).

For the range of accelerating voltages possible with your apparatus, measure and record the diameters of the two prominent rings. Make use of the symmetry of the diffraction pattern to make multiple measurements of the diameters to obtain an estimate of your experimental uncertainty.

It should be noted that the diffraction rings on the phosphor screen are rather broad. Fig. 9 shows one possible contributing factor to the breadth of the rings.

{ ${/download/attachments/164078099/Screen%20Geometry.png?version=1&modificationDate=1565991177000&api=v2}$ Figure 9: Geometry of the diffracted electrons at the phosphor screen.

The electrons satisfying the Bragg condition are diffracted into a cone. The intersection of this cone with the phosphor screen forms a hyperbola. Since our sample consists of randomly orientated micro-crystals, the ring pattern we observe consists of many such hyperbolas smeared about the tube axis. Thus, the most accurate measure of θ should be at the bottom of the hyperbola (the small radius side for each ring).

From the diameters, deduce the diffraction angles { $\theta$ { $\theta$ , for each accelerating potential and for each lattice spacing. Use the Bragg condition for constructive interference, Eq. (4), to calculate the electron wavelength for each accelerating potential. Using appropriate propagation of uncertainties, quote an experimental uncertainty for each wavelength.

From the accelerating potential for each measurement, calculate the electron’s kinetic energy. Is it necessary to account for relativistic effects when calculating the momentum? What is the uncertainty in your measurement of the electron kinetic energy?

Plot { $\lambda$ { $\lambda$  versus { $1/p$ { $1/p$  for the electron. Extract a value for Planck's constant from the plot. Are your data consistent with de Broglie's relation, Eq. (2), to within experimental uncertainties?

In the early 1950s, one of the most challenging questions in biophysics involved the structure of DNA. Rosalind Franklin was studying macroscopic fibers of DNA, that is, highly condensed phases of DNA molecules forming a columnar phase. In a columnar phase there is no order along the axis of the fiber. In the plane of the fiber, perpendicular to the axis, the DNA molecules are organized in a hexagonal lattice as in Fig. 10.

{ ${/download/attachments/164078099/DNA%20Fibers.png?version=1&modificationDate=1570038482000&api=v2}$ Figure 10: A bundle of DNA fibers.

In order to determine the structure of DNA, Franklin performed x-ray diffraction experiments on such fibers. The x-ray beam was perpendicular to the axis of the fiber and diffraction patterns were recorded. One of Franklin’s famous x-ray diffraction photographs of DNA is shown in Fig. 11.

{ ${/download/attachments/164078099/Fig_11.png?version=1&modificationDate=1499784721000&api=v2}$ Figure 11: X-ray diffraction pattern of DNA molecules.

In 1953 James Watson and Francis Crick used the x-ray diffraction images produced by Rosalind Franklin to determine the structure of DNA. For this work, Watson and Crick were awarded the Nobel Prize in 1962. We use a model here to help you understand what they did. The goal is two-fold:

• We wish to explore the relationship between an object and its diffraction pattern, and
• We shall apply this relationship to study the structure of DNA.

It is difficult to produce crystals of aligned DNA molecules in our lab. Therefore, we will use a model of DNA, containing its essential geometrical features. Given the dimensions of the repeating features of DNA, x-rays have appropriate wavelengths to produce a diffraction pattern. Since it is desirable to see the diffraction pattern, we will use visible light from a HeNe laser and an enlarged model of DNA to produce a diffraction pattern at this longer wavelength. The model consists of black markings on a 35 mm slide. An enlarged map of the slide is shown in Fig. 12.

{ ${/download/attachments/164078099/ICE.png?version=1&modificationDate=1499784594000&api=v2}$ Figure 12: DNA Model slide (Institute for Chemical Education, University of Wisconsin, Department of Chemistry).

Shine a laser through the patterns and project the diffraction pattern onto a screen or wall. Start with pattern A and move as the arrows suggest. For each pattern, make a sketch of the diffraction pattern in your lab notebook. The most complete model of DNA is given in pattern J. With each of your sketches, explain which features of DNA contribute to which parts of the diffraction pattern.