# Atomic and Nuclear Physics

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1 Atomic and Nuclear Physics X-ray physics Physics of the atomic shell LEYBOLD Physics Leaflets Bragg reflection: diffraction of x-rays at a monocrystal P Objects of the experiment To investigate Bragg reflection at an NaCl monocrystal using the characteristic x-ray radiation of molybdenum. To determine the wavelength for the characteristic K and K β x-ray radiation of molybdenum. To confirm Bragg s law of reflection. To verify the wave nature of x-rays Ste Principles In 1913, H. W. and W. L. Bragg realized that the regular arrangement of atoms and/or ions in a crystal can be understood as an array of lattice elements on parallel lattice planes. When we expose such a crystal to parallel x-rays, additionally assuming that these have a wave nature, then each element in a lattice plane acts as a scattering point, at which a spherical wavelet forms. According to Huygens, these spherical wavelets are superposed to create a reflected wavefront. In this model, the wavelength remains unchanged with respect to the incident wave front, and the radiation directions which are perpendicular to the two wave fronts fulfill the condition angle of incidence = angle of reflection. Constructive interference arises in the rays reflected at the individual lattice planes when their path differences are integral multiples of the wavelength. = n with n = 1, 2, 3, (I) As Fig. 1 shows for two adjacent lattice planes with the spacing d, we can say for the path differences 1 and 2 of the incident and reflected rays with the angle : 1 = 2 = d sin so that the total path difference is = 2 d sin. (II) (I) and (II) give us Bragg s law of reflection: n = 2 d sin (III) The angle is known as the glancing angle. In this experiment, we verify Bragg s law of reflection by investigating the diffraction of x-rays at an NaCl monocrystal in which the lattice planes are parallel to the cubic surfaces of the unit cells of the crystal. The lattice spacing d of the cubic 1

2 P LEYBOLD Physics Leaflets Apparatus 1 X-ray apparatus End-window counter for, β, and x-ray radiation additionally required: 1 PC with Windows 9x or Windows NT face-centered NaCl crystal is half the lattice constant a 0. We can thus say [1] 2 d = a 0 = The measurements are conducted using the built-in goniometer of the x-ray apparatus ( ). The x-rays are detected using a GM counter tube (end-window counter) which is swiveled in tandem with the NaCl crystal in a 2_ coupling with respect to the incident light; this means that the counter tube always advances by an angle which is twice that of the crystal (cf. Fig. 2). The x-ray radiation consists of the bremsstrahlung continuum and several sharply defined lines which correspond to the characteristic x-ray radiation of the Mo anode and which originate in the K and K transitions of the molybdenum atoms. This characteristic radiation is particularly suitable for investigating Bragg s law. Its properties are known from the literature [2] and summarized in table 1. Table 2 shows the corresponding glancing angles at which the diffraction maxima of the characteristic radiation are to be expected for scattering at an NaCl monocrystal (d = ) up to the third diffraction order. Fig. 1 Diagram of the reflection of x-rays at the lattice planes of a monocrystal. 1, 2 : path differences, : glancing angle, d: spacing of lattice planes Table 1: Energy E, frequency and wavelength of the characteristic x-ray radiation of molybdenum (weighted mean values [1]) E kev EHz K K kev = 10 3 ev, EHz = Hz, = m Table 2: Glancing angle of the characteristic x-ray radiation of molybdenum for diffraction at an NaCl monocrystal up to the third order Safety notes The x-ray apparatus fulfills all German regulations governing an x-ray apparatus and fully protected device for instructional use and is type approved for school use in Germany (NW 807/97 Rö). The built-in protection and screening measures reduce the local dose rate outside of the x-ray apparatus to less than 1 Sv/h, a value which is on the order of magnitude of the natural background radiation. Before putting the x-ray apparatus into operation, inspect it for damage and check to make sure that the high voltage shuts off when the sliding doors are opened (see Instruction Sheet of x-ray apparatus). Keep the x-ray apparatus secure from access by unauthorized persons. Do not allow the anode of the x-ray tube Mo to overheat. When switching on the x-ray apparatus, check to make sure that the ventilator in the tube chamber is turning. The goniometer is positioned solely by electric stepper motors. Do not block the target arm and sensor arm of the goniometer and do not use force to move them. Fig. 2 n (K ) (K ) Diagram showing the principle of diffraction of x-rays at a monocrystal and 2 coupling between counter-tube angle and scattering angle (glancing angle) 1 collimator, 2 monocrystal, 3 counter tube 2

4 P LEYBOLD Physics Leaflets Fig. 4 Diffraction spectrum of x-ray radiation for Bragg reflection to the third order at an NaCl monocrystal Top: linear representation of counting rate R Bottom: logarithmic representation of counting rate R Parameters of x-ray tube: U = 35 kv and I = 1 ma 4

5 LEYBOLD Physics Leaflets P Evaluation Access the evaluation functions of the software X-ray Apparatus by clicking the right-hand mouse button and select the command Calculate Peak Center. Using the left mouse button, mark the entire width of the peaks; if desired, insert the calculated peak center and the peak width in the diagram with Alt+T and note the center as the glancing angle in the measurement table (see tables 3 and 4). Save your measurements and evaluations to a suitably named file with the button or by pressing F2. Using the glancing angle and the lattice plane spacing d = , calculate the wavelength using Bragg s law of reflection (IV) (see tables 3 and 4). Find the mean values for the individual diffraction orders of the measured wavelengths (see table 5). Table 3: Measured glancing angles of the Mo K line and the calculated wavelengths for the first through third diffraction orders Table 5: Mean value and literature value [2] for the characteristic wavelength (K ) (K ) Mean value Literature value Results The close agreement of the experimentally determined wavelengths for the characteristic lines with the literature values in table 5 verify the validity of Bragg s law. This simultaneously confirms the wave nature of x-rays, as this property was assumed in the process of deducing this law. n (K ) (K ) Additional information The characteristic K and K lines actually consist of multiple, adjacent discrete lines, which can be observed separately at higher diffraction orders (see Physics Leaflet P ). Table 1 shows the weighted mean values of the respective individual lines from this substructure. Table 4: Measured glancing angles of the Mo K line and the calculated wavelengths for the first through third diffraction orders n (K ) (K ) Literature [1] Handbook of Chemistry and Physics, 52nd Edition ( ), The Chemical Rubber Company, Cleveland, Ohio, USA. [2] C. M. Lederer and V. S. Shirley, Table of Isotopes, 7th Edition, 1978, John Wiley & Sons, Inc., New York, USA. LEYBOLD DIDACTIC GMBH Leyboldstrasse 1 D Hürth Phone (02233) Telefax (02233) Telex LHPCGN D by Leybold Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved

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