Electron Diffraction ==================== Background ---------- In 1923, in his doctoral dissertation, Louis de Broglie proposed that all forms of matter have wave as well as particle properties, just like light. The wavelength, :math:`\lambda`, of a particle, such as an electron, is related to its momentum, :math:`p`, by the same relationship as for a photon: .. math:: :label: wl \lambda = h/p where :math:`h` is Planck's constant. The wave properties of electrons are illustrated in this experiment by the interference, which results when they are scattered from successive planes of atoms in a target composed of graphite micro crystals. The spacing between successive planes is obtainable from the interference pattern. .. _ed_fig1: .. figure:: pictures/bragg_reflection_simple.png :align: center :alt: Simple grid with possible atomic planes which reflect incoming X-rays leading to diffraction. Total path-length difference Delta, between 2 planes. :scale: 30 % Reflection of electron waves from atomic planes. Ray 1 and 2 have no path length difference, :math:`2 \Delta` is the path length difference between ray 1 from the top plane and ray 3 from the one below. The short-dashed lines show another possible set of atomic planes. A useful model for the formation of diffraction pattern in X-ray diffraction is due to W.H and W.L Bragg (1913). They regarded the crystal to me made of parallel (atomic) planes from which the X-ray or electron waves are reflected specularly (incident angle equal reflected angle). Only a small fraction of the wave is reflected from each plane and the final superposition of these reflected waves lead to the observed diffraction pattern. For a single crystal, strong reflection of waves occurs when the so-called Bragg condition is met: .. math:: :label: br 2d sin(\theta) = n\lambda\ \ (n = 1.,2, ...) Which is the same as the one for X-ray diffraction (see the explanation there). Here :math:`\lambda` is electron matter wave length, :math:`\theta` is the incident angle and :math:`d` is the distance between the atomic planes. In the graphite target, there are very many perfect micro crystals randomly oriented to one another. .. _ed_fig2: .. figure:: pictures/electron_diffraction_schematic.png :align: center :alt: Schematic of electron diffration, incoming electron beam at angle theta with respect to cristal, scattered by 2 theta into detector. :scale: 40 % Geometry for diffraction from a single graphite crystal Therefore the strongly emerging beam will be of a conical shape of half-angle :math:`2\theta` as shown in :numref:`ed_fig3`. If this beam falls on a phosphor-coated screen, rings of light will be formed. .. _ed_fig3: .. figure:: pictures/electron_diffraction_cone.png :align: center :alt: Cone of scattered electrons at 2 theta around incoming beam starting at the target. :scale: 40 % Diffraction from a large number of micro crystals Experimental Setup ------------------ The apparatus is shown in :numref:`ed_fig4`. Electrons emitted by thermionic emission from a heated filament (4) inside the cathode are accelerated towards the graphite target (9) of the anode by a potential difference, :math:`V_a` between the cathode and anode. A focusing electrode (8) is located in front of the target to focus the electron beam in order to provide a sharp interference pattern on the screen (11). .. _ed_fig4: .. figure:: pictures/electron-diffraction_overview.png :align: center :alt: Schematic of electron diffraction tube, with filament, anode, target and screen at the end of the spherical glass type for viewing. :scale: 40 % Overview of the electron diffraction tube.(1)4-mm socket for filament heating supply, (2) 2-mm socket for cathode connection, (3) internal resistor, (4) filament. (5) cathode, (6) anode, (7) 4-mm plug for anode connection (HV), (8) focusing electrode, (9) polycrystalline graphite grating, (10) Boss, (11) fluorescent screen. Their kinetic energy, :math:`K`, on reaching the target is equal to their loss of potential energy: .. math:: :label: kin K = \frac{p^2}{2m} = eV_a from equations :eq:`wl` and :eq:`kin` one obtains: .. math:: :label: kin1 \lambda = \frac{h}{\sqrt{2 m V_a e}} \lambda(nm) = 1.228/\sqrt(V_a) Experimental Procedure ---------------------- Connect the power supplies, a historic HV voltmeter (for high voltage (HV) measurements) and a micro-ammeter to the vacuum tube as shown in :numref:`ed_fig5` .. _ed_fig5: .. figure:: pictures/electron_diffraction_wiring.png :align: center :alt: Wiring diagram for electron diffraction tube. :scale: 40 % Connecting the electron diffraction tube **HAVE THE CIRCUIT CHECKED BY YOUR INSTRUCTOR BEFORE TURNING ANYTHING ON.** The use the historic volmeter to measure the acceleration voltage Adjust the voltage controls on the power supplies to zero and then turn them on. Wait a few minutes for the filament to warm up. Set the focusing/intensity voltage to around 10 - 15 V and then slowly increase the accelerating voltage to 3 kV. A bright central spot and two rings should be observable on the screen. The rings are due to first order (n = 1) diffraction from two different sets of atomic planes having different spacings. Adjust the focusing/intensity voltage until the rings are as sharply defined as possible and then measure their diameters. Try to find a setting that produces the best images for the lowest accelerating voltage, where the rings are barely visible (about 1.5 kV) , as well as the highest accelerating voltage :math:`V_a` (about 4kV) Obtain measurements for 5 - 6 different accelerating voltages, :math:`V_a`, over as wide a range as possible. Use the IDS camera to take pictures of the rings. Determine the ring diameters and their uncertainties from the images using the `ImageAnalyzer `_. For help ask an instructor. Analysis -------- The geometry for the vacuum tube is shown in :numref:`ed_fig6`. .. _ed_fig6: .. figure:: pictures/electron_diffraction_geometry.png :align: center :alt: Geometry between target and spherical screen surface and scattering angle. :scale: 40 % Vacuum tube geometry The value of :math:`\theta` can be obtained from .. math:: :label: g1 tan(2\theta) = \frac{D/2}{l_1 + l_2} l_1 = L - R l_2 = \sqrt{R^2 - (D/2)^2} here :math:`R = 6.25\ cm` and :math:`L = 13.7\ cm`. The structure of graphite is shown in :numref:`ed_fig7`. The two possible layers are indicated by dashed lines. #. Using the ImageAnalyzer and the calibration image to determine the image scale needed to determine the ring diameters (see `ImageAnalyzer `_ ) #. The most effective way to determine the ring diameters in each image is to determine the location of the ring edges from left to right as shown in :numref:`ed_fig8`. Write a function that returns the two ring diameters and their uncertainties and uses the the data file name as argument. (*hint:* you can calculate derivatives numerically see the `Collection of useful tools `_ #. Write a function that calculates the atomic plane separation including its uncertainty based on equations :eq:`g1`, :eq:`kin1` and :eq:`br`. #. Make a plot of your result as a function of the voltages. Include all error bars. Determine the weighted average values and their uncertainties. #. From your measured data determine the distance :math:`s` between the carbon atoms and compare your result to the value provided in :numref:`ed_fig7`. Does it agree within your error bars, if not how big is the deviation in units of your error bar? .. _ed_fig7: .. figure:: pictures/electron_diffraction_graphite.png :align: center :alt: Hexagonal structure of graphite and the two possible planes with distance d1 and d2. s (0.142 nm) is the distance between atoms. Alpha is 120 degrees, the angle between two bonds. :scale: 40 % The structure of graphite with Bragg planes. .. _ed_fig8: .. figure:: pictures/2_5kV.png :align: center :alt: Image of electron diffraction screen with selected ring edge locations :scale: 70 % Determination of the locations of the diffraction ring edges from left to right. From these data the ring diameters and their uncertainties can be determined. .. include:: include/links.rst