Investigations of Wave-Particle Duality by Kent Ames North Gwinnett High School

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1 Investigations of Wave-Particle Duality by Kent Ames North Gwinnett High School The Problem: To demonstrate to the student how modern analytical techniques used in university and engineering laboratories derive from basic theories with which the student is already familiar. Abstract: Most high school physics students are exposed to very perfunctory labs which examine specific physics concepts but give the student little insight into practical experimental techniques found in research and engineering labs. Students understand many basic concepts of physics such as work, energy, momentum, and wave theory but have no idea about how these phenomena can be utilized in the fields of physics, chemistry, and engineering to analyze structural characteristics, examine atomic detail, and construct modern materials and integrated circuit devices. This lab provides a bridge between phenomena previously examined in the high school lab such as standing waves, resonance, interference, optical diffraction, electronic circuits, and the quantum physics concept of wave-particle duality. At the same time it exposes the student to state-of-the-art experimental techniques used in university research labs and engineering facilities across the world. It is expected that students performing this lab have already covered the concepts and performed labs in the areas discussed above; standing waves on a string, resonance in closed/open tubes (speed of sound), diffraction, circuits (Ohm s law). These labs will familiarize the students with the ideas of quantization in wave behavior, constructive and destructive interference, and meters, wiring, and safety associated with circuits. Alignment with standards: AP College Board AP Physics B Standards (National Standards) III. ELECTRICITY AND MAGNETISM A. Electrostatics 2. Electric field and electric potential (including point charges) b) Students should understand the concept of electric potential, so they can: (2) Calculate the electrical work done on a charge or use conservation of energy to determine the speed of a charge that moves through a specified potential difference. (5) Calculate how much work is required to move a test charge from one location to another in the field of fixed point charges. (6) Calculate the electrostatic potential energy of a system of two or more point charges, and calculate how much work is required to establish the charge system. C. Electric circuits d) Students should understand the properties of voltmeters and ammeters, so they can: (2) Identify or show correct methods of connecting meters into circuits in order to measure voltage or current. IV. WAVES AND OPTICS A. Wave motion (including sound) 1. Traveling waves Students should understand the description of traveling waves, so they can: b) Apply the relation among wavelength, frequency, and velocity for a wave. 4. Superposition Students should understand the principle of superposition, so they can apply it to traveling waves moving in opposite directions B. Physical optics 1. Interference and diffraction Students should understand the interference and diffraction of waves, so they can: a) Apply the principles of interference to coherent sources in order to: (1) Describe the conditions under which the waves reaching an observation point from two or more sources will all interfere constructively, or under which the waves from two sources will interfere destructively. (2) Determine locations of interference maxima or minima for two sources or determine the frequencies or wavelengths that can lead to constructive or destructive interference at a certain point.

2 b) Apply the principles of interference and diffraction to waves that pass through a single or double slit or through a diffraction grating, so they can: (1) Sketch or identify the intensity pattern that results when monochromatic waves pass through a single slit and fall on a distant screen, and describe how this pattern will change if the slit width or the wavelength of the waves is changed. (2) Calculate, for a single-slit pattern, the angles or the positions on a distant screen where the intensity is zero. (3) Sketch or identify the intensity pattern that results when monochromatic waves pass through a double slit, and identify which features of the pattern result from single-slit diffraction and which from two-slit interference. (4) Calculate, for a two-slit interference pattern, the angles or the positions on a distant screen at which intensity maxima or minima occur. (5) Describe or identify the interference pattern formed by a diffraction grating, calculate the location of intensity maxima, and explain qualitatively why a multiple-slit grating is better than a two-slit grating for making accurate determinations of wavelength. c) Apply the principles of interference to light reflected by thin films, so they can: V. ATOMIC AND NUCLEAR PHYSICS A. Atomic physics and quantum effects 1. Photons, the photoelectric effect, Compton scattering, x-rays a) Students should know the properties of photons, so they can: (1) Relate the energy of a photon in joules or electron-volts to its wavelength or frequency. (2) Relate the linear momentum of a photon to its energy or wavelength, and apply linear momentum conservation to simple processes involving the emission, absorption, or reflection of photons. 3. Wave-particle duality Students should understand the concept of de Broglie wavelength, so they can: a) Calculate the wavelength of a particle as a function of its momentum. b) Describe the Davisson-Germer experiment, and explain how it provides evidence for the wave nature of electrons. LABORATORY AND EXPERIMENTAL SITUATIONS These objectives overlay the content objectives, and are assessed in the context of those objectives. 1. Design experiments Students should understand the process of designing experiments, so they can: a) Describe the purpose of an experiment or a problem to be investigated. b) Identify equipment needed and describe how it is to be used. c) Draw a diagram or provide a description of an experimental setup. d) Describe procedures to be used, including controls and measurements to be taken. 2. Observe and measure real phenomena Students should be able to make relevant observations, and be able to take measurements with a variety of instruments (cannot be assessed via paper-and-pencil examinations). 3. Analyze data Students should understand how to analyze data, so they can: a) Display data in graphical or tabular form. b) Fit lines and curves to data points in graphs. c) Perform calculations with data. d) Make extrapolations and interpolations from data. 4. Analyze errors Students should understand measurement and experimental error, so they can: a) Identify sources of error and how they propagate. b) Estimate magnitude and direction of errors. c) Determine significant digits. d) Identify ways to reduce error. 5. Communicate results Students should understand how to summarize and communicate results, so they can: a) Draw inferences and conclusions from experimental data. b) Suggest ways to improve experiment. c) Propose questions for further study.

3 Georgia State Standards (QCC) Co-Requisite Characteristics of Science Habits of Mind SCSh1. Students will evaluate the importance of curiosity, honesty, openness, and skepticism in science. a. Exhibit the above traits in their own scientific activities. b. Recognize that different explanations often can be given for the same evidence. c. Explain that further understanding of scientific problems relies on the design and execution of new experiments which may reinforce or weaken opposing explanations. SCSh2. Students will use standard safety practices for all classroom laboratory and field investigations. a. Follow correct procedures for use of scientific apparatus. b. Demonstrate appropriate technique in all laboratory situations. c. Follow correct protocol for identifying and reporting safety problems and violations. SCSh3. Students will identify and investigate problems scientifically. a. Suggest reasonable hypotheses for identified problems. b. Develop procedures for solving scientific problems. c. Collect, organize and record appropriate data. d. Graphically compare and analyze data points and/or summary statistics. e. Develop reasonable conclusions based on data collected. f. Evaluate whether conclusions are reasonable by reviewing the process and checking against other available information. SCSh4. Students will use tools and instruments for observing, measuring, and manipulating scientific equipment and materials. a. Develop and use systematic procedures for recording and organizing information. b. Use technology to produce tables and graphs. c. Use technology to develop, test, and revise experimental or mathematical models. SCSh5. Students will demonstrate the computation and estimation skills necessary for analyzing data and developing reasonable scientific explanations. a. Trace the source on any large disparity between estimated and calculated answers to problems. b. Consider possible effects of measurement errors on calculations. c. Recognize the relationship between accuracy and precision. d. Express appropriate numbers of significant figures for calculated data, using scientific notation where appropriate. e. Solve scientific problems by substituting quantitative values, using dimensional analysis and/or simple algebraic formulas as appropriate. SCSh6. Students will communicate scientific investigations and information clearly. a. Write clear, coherent laboratory reports related to scientific investigations. b. Write clear, coherent accounts of current scientific issues, including possible alternative interpretations of the data c. Use data as evidence to support scientific arguments and claims in written or oral presentations. d. Participate in group discussions of scientific investigation and current scientific issues. The Nature of Science SCSh7. Students will analyze how scientific knowledge is developed. Students will recognize that: b. Universal principles are discovered through observation and experimental verification. c. From time to time, major shifts occur in the scientific view of how the world works. More often, however, the changes that take place in the body of scientific knowledge are small modifications of prior knowledge. Major shifts in scientific views typically occur after the observation of a new phenomenon or an insightful interpretation of existing data by an individual or research group. d. Hypotheses often cause scientists to develop new experiments that produce additional data. e. Testing, revising, and occasionally rejecting new and old theories never ends. SCSh8. Students will understand important features of the process of scientific inquiry. Students will apply the following to inquiry learning practices:

4 a. Scientific investigators control the conditions of their experiments in order to produce valuable data. b. Scientific researchers are expected to critically assess the quality of data including possible sources of bias in their investigations hypotheses, observations, data analyses, and interpretations. d. The merit of a new theory is judged by how well scientific data are explained by the new theory. e. The ultimate goal of science is to develop an understanding of the natural universe which is free of biases. SCSh9. Students will enhance reading in all curriculum areas by: a. Reading in All Curriculum Areas Read both informational and fictional texts in a variety of genres and modes of discourse Read technical texts related to various subject areas b. Discussing books Recognize the features of disciplinary texts. c. Building vocabulary knowledge Demonstrate an understanding of contextual vocabulary in various subjects. Use content vocabulary in writing and speaking. Explore understanding of new words found in subject area texts. d. Establishing context Explore life experiences related to subject area content. Discuss in both writing and speaking how certain words are subject area related. Determine strategies for finding content and contextual meaning for unknown words. SP1. Students will evaluate the forms and transformations of energy. a. Analyze, evaluate, and apply the principle of conservation of energy and measure the components of work-energy theorem by describing total energy in a closed system. identifying different types of potential energy. calculating kinetic energy given mass and velocity. relating transformations between potential and kinetic energy. a. Measure and calculate the vector nature of momentum. b. Compare and contrast elastic and inelastic collisions. c. Demonstrate the factors required to produce a change in momentum. d. Analyze the relationship between temperature, internal energy, and work done in a physical system. SP4. Students will analyze the properties and applications of waves. a. Explain the processes that results in the production and energy transfer of electromagnetic waves. b. Experimentally determine the behavior of waves in various media in terms of reflection, refraction, and diffraction of waves. c. Explain the relationship between the phenomena of interference and the principle of superposition. d. Demonstrate the transfer of energy through different mediums by mechanical waves. SP5. Students will evaluate relationships between electrical and magnetic forces. a. Describe the transformation of mechanical energy into electrical energy and the transmission of electrical energy. b. Determine the relationship among potential difference, current, and resistance in a direct current circuit. c. Determine equivalent resistances in series and parallel circuits. d. Determine the relationship between moving electric charges and magnetic fields.

5 Gwinnett County AKS design and conduct scientific investigations (GPS, HSGT, ACT) (SCPH_A2005-1) 1a - identify, develop and investigate questions/problems that can be answered through scientific inquiry, 1b1 - recognize hypotheses often lead to the development of new experiments (GPS), 1c - develop procedures for solving scientific problems (GPS), 1c1 - control the conditions of scientific investigations (GPS), 1d - collect, organize and record appropriate data (GPS), 1e - recognize different explanations may be given for the same evidence (GPS), 1f - explain further understanding of scientific problems relies on the design and execution of new experiments may reinforce or weaken explanations (GPS), 1f1 - recognize testing, revising, and occasionally rejecting new and existing theories is a continuous process (GPS), 1f2 - recognize universal principles are discovered through observation and experimental verification and basic principles are the same everywhere (i.e., law of conservation of matter) (GPS), 1f3 - recognize major shifts in scientific views typically occur after the observation of a new phenomenon or the interpretation of existing data (GPS), 1g - examine the role of curiosity and skepticism in scientific investigations (GPS), 1h - recognize science disciplines differ from one another in what is studied, techniques used, and outcomes sought (GPS) apply standard safety practices for all classroom laboratory and field investigations (GPS, HSGT) (SCPH_A a - follow correct procedures for use of scientific apparatus (GPS), 2b - demonstrate appropriate techniques in all laboratory situations (GPS), 2c - follow correct protocol for identifying and reporting safety problems and violations (GPS) use technology to collect, observe, measure and manipulate data and findings (GPS, HSGT) (SCPH_A2005-3) 3a - develop and use systematic procedures for recording and organizing information (GPS), 3b - use graphical analysis software to produce tables/graphs and to determine constants in experiments (i.e., the acceleration of gravity) (GPS), 3c - use technology to develop, test and revise experimental/mathematical models (GPS) use valid critical assumptions to draw conclusions (GPS, HSGT, ACT) (SCPH_A2005-4) 4a - develop reasonable conclusions based on data collected (GPS), 4b - evaluate whether conclusions are reasonable by reviewing the process and checking against other available information (GPS), 4c - assess the quality of data critically for possible sources of bias (GPS), 4e - recognize the merit of a new theory is judged by how well scientific data are explained by the new theory (GPS) apply computation and estimation skills necessary for analyzing data and developing conclusions (GPS, HSGT, ACT) (SCPH_A2005-5) 5a - determine the source of large disparities between estimated and calculated results (GPS), 5b - examine the possible effects of measurement errors on calculations (GPS), 5b1 - relate number of significant figures to precision of measuring instrument, 5c - explain the relationship between accuracy and precision (GPS), 5d - express appropriate number of significant figures for calculated data, using scientific notation where appropriate (GPS), 5e - solve scientific problems by substituting quantitative values, using dimensional analysis and/or simple algebraic functions as appropriate (GPS), 5f - compare and analyze data points graphically and/or summary statistics (GPS) communicate scientific investigations clearly (GPS, HSGT) (SCPH_A2005-6) 6a - write clear, coherent laboratory reports related to scientific investigations (GPS), 6b - write clear, coherent accounts of current scientific issues, including possible alternative interpretations of the data (GPS), 6c - use data as evidence to support scientific arguments and claims in written or oral presentations (GPS), 6d - participate in group discussions of scientific investigations and current scientific issues (GPS), 6e - use peer reviews to analyze accuracy of scientific writings/reports (GPS) read scientific materials to establish context for subject matter, develop vocabulary and to be aware of current research (GPS, HSGT) (SCPH_A2005-7) 7a - read grade-level appropriate text (both informational and fictional) from a variety of genres and modes of discourse (GPS)

6 , 7a1 - read technical text related to various subject areas (GPS), 7b - discuss messages and themes from text and relate to other subject areas (GPS), 7b1 - respond to text using multiple modes of discourse (i.e., debate), 7b2 - evaluate the merit of texts, 7b3 - examine the author s purpose in writing, 7b4 - examine the features of disciplinary texts, 7c - use content vocabulary in writing and speaking (GPS), 7d - apply strategies for determining content and contextual meaning for unknown words (GPS), 7e - examine relationship between life experiences and subject area content (GPS) apply mathematical skills and processes to analyze and solve scientific problems (GPS, HSGT) (SCPH_B2005-8) 8a - explain the relationship between fundamental and derived units and give examples of each, 8b - compare and contrast scalar and vector quantities and give examples of each (GPS), 8c - use to-scale vector diagrams to show magnitude and direction, 8d - use vector addition to solve problems for vectors that are on the same line, perpendicular to each other, and not perpendicular to each other (CP, Honors/Gifted only), 8e - use interface technology to plot and interpret the significance of slope, intercepts, and area under a graph, 8f - use statistical graphical analysis software to produce and calculate the slope of a best-fit line, 8f1 - calculate the relative error, 8f2 - analyze sources of error, 8f3 - use slope of line to develop a mathematical model, 8g - suggest how to manipulate data to produce a linear graph given the relationship between variables (Honors/Gifted only) analyze the properties of waves (GPS, HSGT) (SCPH_E ) 23a - describe waves as a means of transporting energy (GPS), 23c - compare and contrast mechanical and electromagnetic waves (GPS), 23d - compare and contrast longitudinal and transverse waves (GPS), 23e - explain the relationship between wavelength, frequency, and wave speed (GPS), 23f - demonstrate and explain the general wave properties of reflection, refraction, interference, and diffraction (GPS), 23g - explain the relationship between the phenomena of interference and the principle of superposition (GPS), 23h - demonstrate and explain wave phenomena using various types of equipment (i.e., ripple tank, slinky, soft rope, signal generator, and oscilloscope) (GPS) analyze the properties of light and optics (GPS, HSGT) (SCPH_E ) 25a - explain the relationship between energy, frequency, wavelength and velocity for all parts of the electromagnetic spectrum (GPS), 25h - demonstrate interference and diffraction effects in a single slit, a double slit, and/or a multiple slit diffraction grating and thin film (include the relationship between spectra and atomic structure) (GPS), 25m - construct ray diagrams and make calculations relating to focal length, image distance, object distance, and image magnification for curved mirrors and lenses.

7 Objective: In this experiment, you will observe the diffraction produced by electrons, calculate the wavelength of the electrons given the production of this interference pattern, and examine the crystalline structure of graphite based on your observations. Anticipated Learner Outcomes: Student will recognize the wave nature of particles Student will apply electric circuit concepts; circuit diagrams, applications of multi-meter Student will calculate the kinetic energy of an electron using potential Student will calculate de Broglie wavelength of an electron Student will observe and measure characteristics of diffraction patterns Student will calculate the atomic spacings of graphite crystal based on Bragg s law Background: Introduction: When an electron beam is made incident upon a thin crystalline sample diffraction rings will appear. The spacing and order of the atoms in the crystal act like slits in a diffraction grating. Electrons passing through the crystal produce constructive and destructive interference resulting in observable patterns on a fluorescent screen. Theory: That matter can have wave particle duality was suggested by the French Scientist Louis de Broglie in the year It was established from Young s experiments on double slit diffraction and interference, and Einstein s theory of the photoelectric effect that light can have wave particle duality, that is that electromagnetic waves can exhibit characteristics of particles. De Broglie then made the bold suggestion that matter too, like light can exhibit dual nature, that particles might exhibit classical particle behavior and possess wave-like properties. This was indeed a bold suggestion considering that there was no evidence at that time to show the wave nature of particles like electrons. According to Bohr s quantum condition, the permitted electron orbits are where the electron s angular momentum (L) has to be an integral multiple of ħ. De Broglie pointed out that this condition was equivalent to that of standing waves in a circular orbit where the circumference of the circle must be an integral multiple of wavelength (λ). De Broglie showed that by using the two equations L = m v r = nħ and 2π r = nλ and by using the momentum (p) of the electron, that the wavelength of the electron wave can be defined by the equation λ = h/p (de Broglie Equation). Considering electron to be waves, it can be shown that the wavelength of a typical electron beam with energy of 100 ev is Å. It is not surprising that the wave properties of matter like electrons were not readily observed because it would require a diffraction grating with width of the order of m to observe diffraction effects. Fortunately, crystals have atomic spacing of an order of meter that they can serve as good diffraction grating for an electron beam. The first confirmation of de Broglie s hypothesis came in 1927 when George Paget Thomson, son of J.J. Thomson, performed an experiment in which electrons Fig. 1 Diffraction pattern from polycrystallines were fired into a thin sheet of metal. He observed ringed diffraction patterns normally associated with diffraction electromagnetic waves in disordered crystals or polycrystallines. He observed that by changing the electron beam s incident energy, the diameters of the diffraction rings change proportionally, as expected from Bragg's equation. In 1913 William Bragg postulated that X-rays would diffract from metals due to the discovery that these metals were crystalline. He found that in these crystals, specific wavelengths and incident angles produce constructive and destructive interference from the crystal structure much like that produced by light passing through a diffraction grating. The constructive regions correspond to intense

8 peaks of reflected radiation (known as Bragg peaks). Bragg defined the separation of the crystal planes (d) then looked at the requirements for an incident wavelength (λ) to be diffracted and result in constructive interference, namely that the reflected Fig. 2 Bragg diffraction from layers of crystal waves must be some integral multiple (n) of the wavelength. In order for this to happen the interfering waves must have a path difference of dsinθ, thus Bragg s diffraction equation for a crystal (1) 2 d sinθ = nλ When a beam of electrons or photons is made to fall on a crystal target, the incident beam is scattered by the atoms of the crystal in different directions. The interference of the scattered beam in certain directions is constructive, resulting in a strong reflected beam. Bragg showed from simple considerations that the direction for which an intense reflected beam is achieved depends on the wavelength of the incident beam, spacing between atoms in the crystal, and the angle at which the beam is incident on the crystal. Bragg considered the regular arrangement of atoms in a crystal to act like a diffraction grating, the width of which is equal to the spacing between such regular and repeating lattice planes. It can be shown from the fig. 6 that the path difference between the waves scattered off the first two planes is equal to 2d sinθ where d is the spacing between planes. If this path difference is equal to an integral multiple of wavelength (λ), constructive interference takes place resulting in a strong beam. Bragg s law is defined by 2d sin θ = n λ Fig. 3 Bragg s law for constructive waves Electron Diffraction and X- ray diffraction Since electrons have a wavelength that is in the order of lattice spacing of the crystals, electron diffraction can be used to study unit cell dimensions and space group symmetry in crystals. Electron beams are much easier to focus than X rays. However they can penetrate up to depths of the order of nanometer and their interactions with the lattice reduces the intensity of the reflected beam. Also the equipment requires a high degree of vacuum. Using thin samples of single crystals, short periods of exposure and limited angular range, diffraction patterns with good resolution can be obtained. Crystal lattices: Any crystalline substance has a regular periodic structure which can be defined in terms of the unit cell. A unit cell is the smallest unit which is representative of the entire crystal symmetry. All crystal structures fit into one of seven basic crystal systems which combined with the various possible lattice centering can be classified into one of the fourteen Bravais lattices. Most crystals belong to the cubic lattice system and are either face centered cubic (fcc) or base centered cubic (bcc). Graphite is an example of a hexagonal crystal lattice. A hexagonal crystal lattice is defined by the set of three geometric axes a 1, a 2 and a 3 as in fig. 4 given at right ( a 3 is the distance between carbon atoms in adjacent planes and is not equal to a 1 or a 2 ). a 1 = a 2 a 3 ; α = β = 90 ; γ = 120 Any lattice point can be defined in terms of these unit vectors as r = ka 1 + l a 2 + ma 3 The crystal planes are generally defined using Miller indices (h k l) which are reciprocals of intersections of planes with the three axes of a unit cell. For example, the Miller indices for the plane in a fcc crystal as in fig 5 is (1 1 1).The distance d between parallel planes with the same set of Miller indices can be Fig 4: coordinate axes and angles in hexagonal crystal Fig 5: (1 1 1) crystal

9 worked out using the equation (2) d = a h + k + l In any crystal there is a family of planes with the same Miller indices, which are parallel to one another. These planes act like mirrors reflecting off the electron beams. However not all planes in a crystal give rise to constructive interference between the reflected beams. In graphite two planes can be identified with inter-planar spacing that satisfy Bragg s condition for constructive interference. The unit cell of graphite is prism shaped. Since the positions of carbon atoms are reproduced in alternate layers, the unit cell includes two layers of atoms. A graphite target therefore should yield two distinctive diffraction patterns. The crystal structure of graphite is shown in Fig.6 below. In each layer each atom has three nearest neighbors, each at a distance of about 1.42 A. The planes of atoms to be considered as reflecting planes in this experiment are the set (100) and the set (111).The perpendicular distance between the sets of parallel planes d 1 and d 2 in graphite are in the ratio : 1. Because of the crystal symmetry there are three sets of rotationally equivalent planes, each at an angle of 120 with respect to the other two sets. Thus the incoming beam is diffracted into a hexagonal array of spots, and the lines connecting the spots intersect at angles of 120. Fig 6: lattice planes (a) and their spacing (d). Diffraction Pattern A single crystal sample produces a diffraction pattern consisting of discrete spots. However most samples are polycrystalline, a substance that is an aggregate of single crystals all randomly oriented. Each of these single crystals produces its own diffraction spots. These spots are so spread out due to the random orientation of the crystals resulting in diffraction rings (Fig 7). A single crystal of graphite should produce a hexagonal array of spots and one for each inter-planar distance d 1 and d 2. However, the graphite sample used in this experiment is powdered graphite which is an aggregate of single crystals. Each crystal produces its own diffraction spots and hence the diffraction pattern is a ring rather than six discrete spots. It can be understood that even though Bragg s condition is satisfied for only particular values of the azimuth angle, it is true for different polar angles, so the spots so formed blur into a ring. In the case of graphite, there should be two concentric rings observed corresponding to the two lattice spacing d 1 and d 2. The central bright spot corresponds to the undiffracted beam. Fig 7: Diffraction rings Wavelength of electron beam The electron beam is accelerated using a high voltage which can be varied between 2 to 5 kv. The kinetic energy of the electron beam can be calculated from the voltage using the work-energy theorem 1 (3) mv 2 = qv 2 Where m and q represent the respective mass and charge of the electron accelerated from rest across a potential (V).

10 Then we use de Broglie s wave equation to determine the wavelength of the electron. (4) λ = h p Since the voltages are not more than 5 kv in this experiment, the speed of the electron is not more than 20% of the speed of light, so it is safe to ignore relativistic effects on the electron. Materials & Supplies: Debye-Scherrer Electron Diffraction Tube (Leybold electron diffraction apparatus) Electron beam high voltage variable power supply ( V) Heater filament Variac power supply (0-10 V a.c.) Focus grid low power supply (0-50 V) Three (3) multi-meters; two used as voltmeters, one as milli-ammeter. 10k potentiometer. Measurement tape or Vernier caliper Hook-up wires Flashlight Experimental setup: The cathode is the filament heated by the 120 V ac supply. The electrons emitted from the cathode are accelerated by the high voltage supply (be careful not to exceed 5000 V). The electron gun arrangement is in the neck of the tube and the graphite target is close to the neck of the tube. The focusing voltage provides the adjustment to get the electron beam in focus. The phosphorescent screen enables to see the fluorescent rings when the electron beam hits the glass wall of the tube. The tube is in high vacuum, so that the electrons do not lose energy by striking the gas atoms. Precautions and Start up: 1. High voltage can be fatal. All connections should be double checked and no bare conductor should be exposed. Use electrical tape where necessary to cover exposed wiring. 2. The glass diffraction tube will implode with undue stress on it, so be careful with calipers when measuring its dimensions. 3. Make sure the anode current does not exceed 0.2 ma, because a highly energetic electron beam can burn a hole in the target and excessive heat on the cathode can reduce its lifetime. 4. It is recommended that the instructor wire up the apparatus due to the presence of high voltage. Have the student check the circuit and receive instructor approval before allowing high voltage to be turned on.

11 Procedures: Startup 1. Read all instructions before beginning the lab. 2. High voltage (V A ) off and set to zero. 3. Measure diameter of electron tube (D), calculate the radius (R) of the tube. 4. Measure distance from graphite crystal to back of electron tube fluorescent screen (l). 5. Turn on focus supply; adjust focus voltage (V F ) to 25 V. 6. Adjust Variac heater (emission) voltage (V C ) to 6 V (maximum is 7 V). 7. Wait one (1) minute for cathode to heat. 8. Turn on high voltage (V A ) ; adjust to 2000 V; note there is long time lag associated with high voltage supply so raise voltage slowly and monitor voltage behavior on multi-meter. 9. Observe electron beam dot at center rear of electron tube on fluorescent screen. Warnings 10. Constantly monitor the anode current (I C ), it should never exceed 0.2 milliamps. As high voltage is increased the anode current will also increase. Adjust the anode current using the focus voltage and heater voltage. Voltages above 4000 V will probably produce better focused patterns if the anode current is kept within a range of 0.01 ma 0.04 ma. 11. Additionally monitor the appearance of the crystal sample within the electron tube. The electron beam will result in heating of the sample, but it should not glow. If it appears to begin glowing with a reddish tint, lower the high voltage, make sure the anode current is below 0.2 ma and consult with instructor.

12 Experimental procedures 12. Set (high) accelerating voltage (V A ) to 2000 V, adjust anode current (I C ) so that it is below 0.2 ma using potentiometer. 13. Turn out room lights and observe pattern projected on back of tube. There should be a central dot, and two rings. Adjust sharpness of rings using focus control voltage (V F ). 14. Measure the arc length (2s) of each ring. 15. Record V A, V F, I C, V C, and inner and outer arc lengths (s i, s o ). 16. Alter high voltage by 50 volt increments and repeat measurements. Your data should span a high voltage range between 1500 and 5000 volts. Starting at 2000 V work your way down to 1500 V, then reverse trend, moving from 2000 V up to 5000 V. Calculations: Use the geometry of the tube shown above to figure out the diffraction angles α and θ for each set of data by using the equations given below: Calculate the speed (v) of the electron for each accelerating voltage using the work-energy relationship (3). Calculate the wavelength (λ) of the electron for each accelerating voltage using de Broglie s wave equation (4). Combining equation (1), (3), and (4) and assuming 1 st order (n=1) yields the following relationship sin θ = d h 1 8me V A

13 Plot a graph of sin θ versus 1 V A for each data table. Draw the best fit line and calculate the slope of the line. Use the slope to calculate the distance d 1 and d 2. These distances represent the diffraction separation distance, but also represent the spacing of atoms in the crystal sample. d = h ( slope 8me ) Using the values of the inter-planar spacing d 1 and d 2 arrive at the value of the closest spacing between atoms ( a 0 ) for the hexagonal graphite lattice using the equations d 1 = 3a 0 / 2 and d 2 = a 0 3/2 Compare your answer for the atomic separation with the published value of graphite. The ratio of the d spacings allows you to judge whether the graphite lattice is hexagonal or cubic since d 1 /d 2 is 3 or 2 for hexagonal or cubic lattices, respectively. For each accelerating voltage determine the ratio of the two d s and determine the average value for this ratio, then determine whether you are looking at a hexagonal or cubic lattice. Data and Analysis: Tube radius (R): Crystal to screen distance (l): Inner Ring Data V C I C V F V A S i α i θ i v λ sinθ i 2 V 1 A Outer Ring Data V C I C V F V A S i α i θ i v λ sinθ i 2 V 1 A Ring Slope d (Å) Inner (d 1 ) Outer (d 2 )

14 Assessment: 1. On a separate calculation sheet, show one example calculation for any value calculated in this lab. 2. Draw a detailed diagram of the equipment and wiring of the electron-diffraction apparatus. 3. Determine R and l for the electron tube. 4. Complete the data tables. 5. Specifically, for V A = 3000 V describe in detail how you focused the images of the rings and how you measured the arc length of the rings. 6. Explain qualitatively what happens to the radius of the rings, and why this happens as the accelerating voltage is increased? (look at your equations) 7. Construct the graphs based on the inner and outer ring data. 8. Determine the slope of each graph. 9. Calculate the inter-planar spacing for each ring based on the slope. 10. Calculate the atomic separations of the graphite crystal. 11. Which diffraction ring represented the smaller atomic spacing? 12. What is the standard (accepted) values for the inter-atomic spacings of graphite (state source)? 13. Calculate the percentage error between your values for the atomic spacings and the standard spacing values. 14. Analyze your results and postulate whether the crystal lattice is hexagonal or cubic. 15. Find and attach a 3D image of the appropriate crystal structure. Error Analysis: Calculate the relative errors listed below: (Δx refers to the precision of your measuring device) ΔR R Δl l ΔS S 19. ΔV V A A 20. Which of these errors is most significant and why?

15 The Plan: The students will perform this lab in groups of three, it is but one lab in a two week long series of advanced labs I hope to introduce following the conclusion of AP exams. The students in the class will review their prior lab work on standing waves, resonance, and optical diffraction as the instructor demonstrates the wave interference previously observed by the students earlier in the semester. The advanced labs including the electron diffraction lab are demonstrated in class to allow the students to see the expected phenomena. The students will record all observations and personal notes in their lab notebooks for future review as the week unfolds. Each group will give a short presentation of their observations and any initial results the day following the lab. In this way the other students who have not done the lab yet can gain valuable insight, and the instructor can ask questions provoking deeper understanding. Summary: At the end of this two-week period students will be asked to do research online to determine how these lab procedures are being used in current research projects. They will be asked to find one example of university or industry research that applies the principles examined in these labs and construct a current events article about that research project. I would also at that time like to bring in several guest engineers or researchers from various research labs (Ga Tech, EMS, Intel, ) do talk about how their labs utilized aspects of these labs the students have been conducting and where that research is leading.