The University of Hong Kong Department of Physics Physics Laboratory PHYS3551 Introductory Solid State Physics Experiment No. 3551-2: Electron and Optical Diffraction Name: University No: This experiment consists of two parts: electron and optical diffraction. Upon finishing these experiments, students should have a good understanding of the wave properties of high energy electrons and diffraction phenomena for electrons and photons. Background reading before this experiment is highly recommended.
Part 1: Electron Diffraction Aim Wave properties of high-energy electrons are displayed by transmitting an electron beam through a film of polycrystalline graphite and observing the diffraction pattern on a screen. The lattice constants of graphite may be obtained from the diameters of the diffraction rings and the accelerating potentials. Background Bragg s law is one of the special case of Laue diffraction. It was first proposed by William Lawrence Bragg and his father William Henry Bragg in 1913, in order to explain their discovery that crystalline solids produced surprising patterns of reflected X-rays. The concept of Bragg diffraction can also applied to neutron diffraction and electron diffraction processes. Bragg s law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. The diffraction of a beam of electrons by atoms or molecules is called electron diffraction. The fact that electrons can be diffracted in a similar way to light and X-rays shows that particles can act as waves. An electron (mass m e, charge e) accelerated through a potential difference V acquires a kinetic energy 1 2 mev ev, (1) 2 where v is the velocity of the electron. The non-relativistic momentum m ev of the electron is 2m e ev. (2) Thus the de Broglie wavelength of an electron is given by h h, (3) m v m ev e 2 e where h is the Planck constant. When a parallel beam of X-rays (electrons in our case) of wavelength strikes a set of crystal (polycrystalline graphite in our case) planes, it is reflected from the different planes and interference occurs between X-rays reflected from adjacent planes (see Fig. 1). Bragg s law states that constructive interference takes place when the difference in pathlength is equal to an integral number of wavelengths: 2dsin n, (4) where n is an integer, d is the spacing between the planes of the atoms (carbon in our case), and (Bragg angle) is the angle between the incident X-ray and the crystal plane. The law enables the structure of many crystals to be determined. Named after W.H. Bragg (1862-1942) and W.L. Bragg (1890-1971).
Incident beam d Reflected beam Figure 1. Bragg s law. The first observation of electron diffraction was by G. Thomson (1892-1975) in 1927, in an experiment in which he passed a beam of electrons in a vacuum through a very thin gold foil onto a photographic plate. Concentric circles were produced by diffraction of electrons by the lattice. The same year C.J. Davisson (1881-1958) and L. Germer (1896-1971) performed a classic experiment in which they obtained diffraction patterns by glancing an electron beam off the surface of a nickel crystal. Both experiments were important verifications of de Broglie s theory and the new quantum theory. Electron diffraction, because of the low penetration, cannot easily be used to investigate crystal structure. It is, however, employed to measure bond lengths and angles of molecules in gases. Moreover, it is extensively used in the study of solid surfaces and absorption. The main techniques are low-energy electron diffraction (LEED) and highenergy electron diffraction (HEED). Experimental: Set-up (see Fig. 2): An electron deflection tube A 0-5 kv high tension power supply with a 6.3 V/2 A filament power supply A variable resistor A multimeter A ruler V (0-5 kv) Deflection tube Ruler ~ 6.3 V/2 A Variable resistor Filament Accelerating system Graphite L r Diffraction ring Figure 2. Set-up. The electron source, the filament electrodes, the accelerating electrodes, the graphite target, and the screen are in an evacuated tube. The tube is evacuated to a pressure of ~10-3 Pa and thus it must be handled carefully to avoid implosion. The high tension involved is about 5 kv. Do not tamper with the unit when high voltages are applied. Before switching on the units, turn the HV (High Voltage) knob to its counter-clockwise
stop position (i.e., at 0 kv), and also set the variable resistance to its maximum. Procedure: The well-defined diffraction rings will appear on the screen after the power supplies are introduced. To determine the diameters of the diffraction rings, measure the inner and the outer edges of the rings with the ruler, and then take averages. Measure the rings corresponding to different voltages ranging from 5.0 kv down to 2.0 kv in steps of 0.2 kv. From Fig. 2, we see that for r being much less than L, r tan2, (5) L where r is the radius of the diffraction ring, and L = 135 mm is the distance between the graphite film and the screen. Combining Eqs. 4 and 5 gives r nl d (6) upon the approximation: tan2 sin2 2sin. For the hexagonal structure of graphite, there are several lattice constants (see Fig. 3). In our measurement, we obtain the interplanar separations d inner (corresponding to r inner) and d outer (corresponding to r outer). Then from Eqs. 3 and 6, d inner and d outer can be obtained from the slopes of the plots of against 1 / V r outer and against 1 / V, respectively. Note the lattice constant a of graphite. r inner and r outer r inner are average radii. Thus obtain d inner a d outer Figure 3. Hexagonal structure of graphite. Error analysis should be included in the calculations. References 1. H.T. Stokes, Solid State Physics (Allyn and Bacon, Boston, 1987).
2. K. Krane, Modern Physics (Wiley, New York, 1996). 3. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996).
Part 2: Optical Diffraction Aim Demonstrate single slit diffraction using a He-Ne laser. Background A slit can be considered as a large number of point sources (Huygens-Fresnel principle), each emitting light spherically in all directions. The amplitude beyond the slit is a result of superposition of wavelets from these point sources. At certain angles, constructive or destructive interference can occur. Destructive interference is obtained at angles sin m /a, m=1,2, (7) where is the wavelength of light, a is the slit size, and L is the distance to screen, as shown in Fig. 1. Fig. 4 illustration of a single slit diffraction. Note that we can distinguish between Fresnel or near-field and Fraunhofer or far-field diffraction. As a rule of thumb, Fraunhofer diffraction will occur at distance L>a 2 /. Experimental: He-Ne laser Apertures (0.5 mm and 0.75 mm) Screen Ruler Procedure Place the 0.5 mm aperture in position, adjust the distance of the screen to obtain clear diffraction rings. Measure the distance to the screen and the position of the rings. Use Eq. (7) to estimate the wavelength of the light for several different values of m. Repeat the procedure for 0.75 mm slit. Discuss the accuracy of the estimate as a function of slit size, distance to screen, order of maximum m, and their experimental uncertainties. Reference: Hecht, E. and Zajac, A., Optics (revised August 2017)