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Atomic and Nuclear Physics Physics LD Atomic and Nuclear Physics Physics Introductory experiments Leaflets P6.1.5.1 Dualism of wave and particle Diffraction of electrons in a polycrystalline lattice (Debye-Scherrer diffraction) Objects of the experiment g Determination of wavelength of the electrons g Verification of the de Broglie’s equation g Determination of lattice plane spacings of graphite Principles Louis de Broglie suggested in 1924 that particles could have wave properties in addition to their familiar particle properties. He hypothesized that the wavelength of the particle is in- versely proportional to its momentum: h λ = (I) p λ: wavelength h: Planck’s constant D1 p: momentum His conjecture was confirmed by the experiments of Clinton Davisson and Lester Germer on the diffraction of electrons at crystalline Nickel structures in 1927. In the present experiment the wave character of electrons is demonstrated by their diffraction at a polycrystalline graphite lattice (Debye-Scherrer diffraction). In contrast to the experi- ment of Davisson and Germer where electron diffraction is observed in reflection this setup uses a transmission diffrac- tion type similar to the one used by G.P. Thomson in 1928. From the electrons emitted by the hot cathode a small beam is singled out through a pin diagram. After passing through a focusing electron-optical system the electrons are incident as sharply limited monochromatic beam on a polycrystalline D graphite foil. The atoms of the graphite can be regarded as a 2 space lattice which acts as a diffracting grating for the elec- trons. On the fluorescent screen appears a diffraction pattern of two concentric rings which are centred around the indif- fracted electron beam (Fig. 1). The diameter of the concentric Fig. 1: Schematic representation of the observed ring pattern due to rings changes with the wavelength λ and thus with the accel- Bi 0206 the diffraction of electrons on graphite. Two rings with diame- erating voltage U as can be seen by the following considera- ters D 1 and D 2 are observed corresponding to the lattice tions: plane spacings d 1 and d 2 (Fig. 3). LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: [email protected] by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved P6.1.5.1 - 2 - LD Physics leaflets From energy equation for the electrons accelerated by the voltage U 1 p2 λ ϑ e ⋅U = m⋅ v2 = (II) n = 2 d sin 2 2⋅m U: accelerating voltage e: electron charge ϑ ϑ m: mass of the particle v: velocity of the particle ∆ ∆ the momentum p can be derived as 1 2 d p = m ⋅ v = 2⋅ e ⋅m ⋅U (III) Substituting equation (III) in equation (I) gives for the wave- length: h λ = (IV) Fig. 2: Schematic representation of the Bragg condition. ⋅ ⋅ ⋅ 2 m e U In 1913, H. W. and W. L. Bragg realized that the regular ar- If we approximate tan 2 ⋅ϑ = sin 2 ⋅ϑ = 2⋅ sin ϑ for small angles rangement of atoms in a single crystal can be understood as we obtain an array of lattice elements on parallel lattice planes. When D we expose such a crystal lattice to monochromatic x-rays or 2⋅ sin ϑ = (VII) ⋅ mono-energetic electrons, and, additionally assuming that 2 L those have a wave nature, then each element in a lattice The substitution of equation (VII) in (V) leads in first order plane acts as a “scattering point”, at which a spherical wave- diffraction (n = 1) to let forms. According to Huygens’ principle, these spherical D wavelets are superposed to create a “reflected” wave front. In λ = d⋅ (VIII) ⋅ this model, the wavelength λ remains unchanged with respect 2 L to the “incident” wave front, and the radiation directions which D: ring diameter are perpendicular to the two wave fronts fulfil the condition L: distance between graphite and screen “angle of incidence = angle of reflection”. Constructive interference arises in the neighbouring rays d: lattice plane spacing reflected at the individual lattice planes when their path differ- ∆ ∆ ∆ ⋅ ⋅ ϑ ences = 1 + 2 = 2 d sin are integer multiples of the wavelength λ (Fig. 2): 2⋅ d⋅sin ϑ = n λ⋅ n = 1, 2, 3, … (V) d: lattice plane spacing ϑ: diffraction angle d1 This is the so called ‘Bragg condition’ and the corresponding diffraction angle ϑ is known as the glancing angle. In this experiment a polycrystalline material is used as diffrac- tion object. This corresponds to a large number of small sin- gle crystallites which are irregularly arranged in space. As a result there are always some crystals where the Bragg condi- d tion is satisfied for a given direction of incidence and wave- 2 length. The reflections produced by these crystallites lie on a cones whose common axis is given by the direction of inci- dence. Concentric circles thus appear on a screen located perpendicularly to this axis. The lattice planes which are im- portant for the electron diffraction pattern obtained with this setup possess the lattice plane spacings (Fig. 3): -10 d1 = 2.13 ⋅10 m -10 d2 = 1.23 ⋅10 m From Fig. 4 we can deduce the relationship Fig. 3 Lattice plane spacings in graphite: ⋅ -10 D d1 = 2.13 10 m tan 2⋅ ϑ = (VI) d = 1.23 ⋅10 -10 m 2 ⋅L 2 LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: [email protected] by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved LD Physics leaflets - 3 - P6.1.5.1 Apparatus 1 Electron diffraction tube......................................555 626 L 1 Tube stand .........................................................555 600 1 High-voltage power supply 10 kV.......................521 70 1 Precision vernier callipers...................................311 54 1 Safety Connection Lead 25 cm red ....................500 611 F 1 Safety Connection Lead 50 cm red ....................500 621 2 2ϑ D 1 Safety Connection Lead 100 cm red ..................500 641 1 Safety Connection Lead 100 cm blue.................500 642 F1 2 Safety Connection Lead 100 cm black ...............500 644 CX A 100 k Ω U λ Fig. 4: Schematic sketch for determining the diffraction angle. Due to equation (IV) the wavelength is determined by the L = 13.5 cm (distance between graphite foil and screen), accelerating voltage U. Combining the equation (IV) and D: diameter of a diffraction ring observed on the screen equation (VIII) shows that the diameters D 1 and D 2 of the ϑ: diffraction angle concentric rings change with the accelerating voltage U: For meaning of F 1, F 2, C, X and A see Fig. 5. 1 D = k ⋅ (IX) U with 2 ⋅L ⋅h Setup k = (X) d⋅ 2 ⋅m⋅ e The experimental setup (wiring diagram) is shown in Fig. 5. Measuring Diameters D 1 and D 2 as function of the accelerat- - Connect the cathode heating sockets F1 and F2 of the ing voltage U allows thus to determine the lattice plane spac- tube stand to the output on the back of the high-voltage ings d 1 and d 2. power supply 10 kV. - Connect the sockets C (cathode cap) and X (focussing electrode) of the tube stand to the negative pole. - Connect the socket A (anode) to the positive pole of the 5 kV/2 mA output of the high-voltage power supply 10 kV. - Ground the positive pole on the high-voltage power supply 10 kV. Safety notes A F F 1 2 When the electron diffraction tube is operated at high vol t- XC F F ages over 5 kV, X-rays are generated. 1 2 g Do not operate the electron diffraction tube with high voltages over 5 keV. 6.3 V~ / 2 A The connection of the electron diffraction tube with grou n- ded anode given in this instruction sheet requires a high - A voltage enduring voltage source for the cathode heating. F , F g Use the high-voltage power supply 10 kV (521 70) for 1 2 C supplying the electron diffraction tube with power. Danger of implosion: the electron diffraction tube is a hig h- vacuum tube made of thin-walled glass. kV 0...max. 0...5V g Do not expose the electron diffraction tube to mechan i- cal stress, and connect it only if it is mounted in the tube 0...5kV 0...5kV --max.100 µA + max.2 mA + stand. g Treat the contact pins in the pin base with care, do not 521 70 bend them, and be careful when inserting them in the tube stand. The electron diffraction tube may be destroyed by voltages Fig. 5: Experimental setup (wiring diagram) for observing the elec- or currents that are too high: tron diffraction on graphite. Pin connection: F1, F 2: sockets for cathode heating g Keep to the operating parameters given in the section C: cathode cap on technical data. X: focusing electrode A: anode (with polycrystalline graphite foil see Fig. 4) LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: [email protected] by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved P6.1.5.1 - 4 - LD Physics leaflets Carrying out the experiment Table 2: Measured diameter D 1 of the concentric diffraction rings as function of the accelerating voltage U. The wave- - Apply an accelerating voltage U ≤ 5 kV and observe the lengths λ1 and λ1,theory are determined by equation (VIII); and diffraction pattern. equation (IV), respectively. Hint: The direction of the electron beam can be influenced by U D λ λ means of a magnet which can be clamped on the neck of 1 1 ,1 theroy tube near the electron focusing system.
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