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ELECTRON DIFFRACTION Ken Cheney 3/21/2006 ABSTRACT We will explore the de Broglie wavelength of particles by diffracting electrons through a carbon film and observing the result on a CRT type screen. PICTURES AND STUDENTS http://www.paccd.cc.ca.us/instadmn/physcidv/physics/teachers/cheney/lab%20manuals/ WEB%20Image%20Folders/Matter%20Waves%20WEB/index.htm NOTATION = Lambda = the wavelength of the particle or wave. h = Plank’s Constant f = frequency v = mu = Frequency n = Integer a = Perpendicular distance between planes of atoms p = Momentum i = The angle between the incoming wave and the normal to the plane = The angle between the plane itself and the incoming wave = The angle of deflection of waves, from the incoming direction to the outgoing direction = 2 L = The distance between the scattering carbon film and the screen D = The diameter of the circles on the screen m = The mass of the electron Va = The voltage that accelerated the electron HISTORY In 1924 Louis de Broglie suggested that particles might have wave properties. The wavelength he proposed was given by: = h/p D:\Docstoc\Working\pdf\c6ad9e59-ec97-49b0-840f-97300a7063f4.DOC 3/13/2011 10:12 PM -1- Recall that light was first shown to be a wave by Young who showed that light interfered with itself, defiantly a property of waves not particles. This suggests trying to demonstrate the wave properties of matter by producing an interference pattern. George Thomson was jointly awarded the Nobel Prize for Physics in 1937 for his work in Aberdeen in discovering the wave-like properties of the electron. The prize was shared with Clinton Joseph Davisson who had made the same discovery independently. Whereas his father had seen the electron as a particle (and won his Nobel Prize in the process), Thomson demonstrated that it could be diffracted like a wave, a discovery proving the principle of wave-particle duality which had first been posited by Louis-Victor de Broglie in the 1920s as what is often dubbed the de Broglie hypothesis. http://en.wikipedia.org/wiki/George_Paget_Thomson Actually Davisson and Germer found this effect for reflection (initially by accident) while Thompson did a transmission experiment. This wave like behavior of particles also led to an inability to say exactly where a particle was or would be, one could only calculate (with fantastic accuracy) the probabilities of finding a particle her or there. This uncertainty made many excellent physicists (such as Einstein) very unhappy! THEORY See “BRAGG SCATTERING WITH MICROWAVES” for the derivation of the equation that gives the direction of maximum scattering from parallel planes of atoms: cos(i)=2a/n or usually in this case sin() = 2a/n Where i is the optimum angle between the normal to the plane and the incoming wave, a is the separation of the reflecting planes, n is an integer, lambda is the wavelength of the wave or particle, and is the angle between the plane and the incoming wave. There are many possible separations between parallel planes in different directions. For example in a cubic crystal some planes are through the faces of the cubes and other planes are diagonally across the cube. We will be interested in the ration of these two interplaner distances so show that for these two choices the ratio is one to the square root of two. D:\Docstoc\Working\pdf\c6ad9e59-ec97-49b0-840f-97300a7063f4.DOC 3/13/2011 10:12 PM -2- We will also be interested in hexagonal shapes as sketched below. Calculate the ratios of the interplaner distances for the planes shown and any others you can devise. B B C C D D B B C C GRAPHITE STRUCTURE Graphite forms in sheets one atom thick. Within the sheets the atoms are arranged in hexagons with nearest neighbors along the side of the hexagon 1.415 Angstroms apart. The sheets are arranged in parallel layers 3.35 Angstroms apart. The sheets are offset so each carbon atom is directly below and above the center of hexagons in the sheets below and above. For a fine tutorial on this see : http://invsee.asu.edu/nmodules/Carbonmod/bonding.html D:\Docstoc\Working\pdf\c6ad9e59-ec97-49b0-840f-97300a7063f4.DOC 3/13/2011 10:12 PM -3- OUR PLAN We want to check: 1. Are electrons (or some charged particles ) producing the circles on the screen? Try a magnet! 2. If it is electrons is there evidence that their wavelength depends on their momentum as predicted by de Broglie? 3. Can we calculate the interatomic spacing from our observations, and is the result reasonable? 4. Do we have evidence of the shape of the crystal? MORE THEORY We want to use the de Broglie wavelength of electrons to predict the relation between the circles the electrons trace on a CRT type screen and the accelerating voltage given the electrons. This analysis should yield the interatomic spacing. According to de Broglie = h/p Kinetic Energy = ½ mv2 = p/2m The energy given the electrons by the accelerating voltage = qVa Combining these: = h / 2mqVa Eq. 1 What we observe The accelerated electrons go through a carbon film, travel a distance L, to a CRT type screen and produce circles of diameter D, they have been deflected through an angle : sin() = D/2L Eq. 2 Out In Plane of atoms The angle of deflection is twice the angle between the incoming wave and the plane (specular reflection) so alpha equals two theta. For small angles : sin( ) sin(2 ) 2sin( ) Continuation of in Eq. 3 And from Bragg D:\Docstoc\Working\pdf\c6ad9e59-ec97-49b0-840f-97300a7063f4.DOC 3/13/2011 10:12 PM -4- sin() = n/2a Eq. 4 Combining equations 1-4 and arranging to look linear: 1/ Va aRD where 1 2mq R 2 L nh Notice that R is a constant so if we plot RD verses 1/ Va we should get a straight line with slope a if de Broglie and Bragg are right! WARNING This is FAR from the complete theory! We can just hope that this simplified version will give some interesting approximations. Once the rings have been seen we know that electrons do indeed have wave properties, the details can be worked out later! EQUIPMENT The tube that does the diffraction has an electron source like a CRT with a heater (filament) heated by 6.3v AC, an anode that boils off the electrons, a shield with feedback through a resistor to prevent too much current (damaging the carbon film), a carbon fill where the diffraction occurs, and a CRT type screen that shows where the electrons go. The carbon is probably a random mix of graphite (ordered sheets of carbon) mixed in surrounding amorphous carbon. The random mix of graphite means that the electrons will strike graphite planes at all possible angles. In our case only two angles will result in useful Bragg angles but these angles occur in all directions around the line of the incoming electrons, hence circular traces on the screen. D:\Docstoc\Working\pdf\c6ad9e59-ec97-49b0-840f-97300a7063f4.DOC 3/13/2011 10:12 PM -5- ELECTRON DIFFRACTION TUBE Connectors Filamen Carbon Cathode Film Electrons t 6.3 v AC Bias Resistor Glass Tube High Voltage Va TO DO Get data and do calculations to check 1-4 in “Our Plan” above. Try several voltages from 2500 to 5000 volts. Plot as suggested above for each circle, two lines and two slopes and two interatomic spacing. Find the ratio of the interatomic spacing and see if the ratio matches the ratio for any planes you can draw through a cubic or hexagonal crystal. The spacing should be of the order of two angstroms, 2 10-10m. D:\Docstoc\Working\pdf\c6ad9e59-ec97-49b0-840f-97300a7063f4.DOC 3/13/2011 10:12 PM -6-

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ELECTRON DIFFRACTION, Chapter 6, Statistical Mechanics, Chapter 17, Dimensional Schr, Radical Eye Software, Take Home, odinger Equation, Nuclear Structure, Fraunhofer diffraction

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