Solid-state physics LEYBOLD Physics Properties of crystals Leaflets P7.1.2.1 X-ray structural analysis Bragg reflection: determining the lattice constants of monocrystals Objects of the experiment Investigating and comparing Bragg reflection at an LiF and an NaCI monocrystal. Determining the lattice constant ao of NaCl and LiF. Principles Bragg's law of reflection describes the diffraction of plane set of lattice planes and is often referred to as the glancing waves at a monocrystal as the selective reflection of the waves angle. at a set of lattice planes within the crystal.Due to the periodicity In a cubic crystal with NaCl structure(cf.Fig.1),the lattice of the crystal,the lattice planes of a set have a fixed spacing planes run parallel to the surfaces of the crystal's unit cells in d.An incident wave with the wavelength A is reflected with maximum intensity when the Bragg condition the simplest case.Their spacing dcorresponds to one half the lattice constant: n.λ=2.d.sin8 n:diffraction order d号 (① λ：wavelength d:spacing of lattice planes This lets us use(I)as an equation for determining the lattice constant ao: is fulfilled (see experiment P6.3.3.1).The angle shows the direction of the incident and reflected wave with respect to the n·入=ao·sin8 (00 In other words,to determine ao we need to measure the glancing angle for a known wavelength A and diffraction Fig.1 Three-dimensional representation of the structure of NaCl order n.This method is more precise when the glancing angles d:Spacing of lattice planes in [1,0,0]-direction are also measured in higher diffraction orders. ao:lattice constant In this experiment,the molybdenum x-rays are used as radia- tion of a known wavelength.Table 1 shows its wavelengths A. Table 1:Wavelengths of the characteristic x-ray radiation of molybdenum(weighted means [1)) Line pm Ka 71.08 KB 63.09 A Geiger-Muller counter tube is used to detect the x-rays;this instrument and the crystal are both pivoted with respect to the e incident x-ray beam in 2 coupling-the counter tube is turned by twice the angle of the crystal(cf.Fig.2).The zero point 0 is characterized by the fact that the lattice planes and theFig. 1 Three-dimensional representation of the structure of NaCl d: Spacing of lattice planes in [1,0,0]-direction a0: lattice constant Solid-state physics Properties of crystals X-ray structural analysis LEYBOLD Physics Leaflets Bragg reflection: determining the lattice constants of monocrystals Objects of the experiment Investigating and comparing Bragg reflection at an LiF and an NaCl monocrystal. Determining the lattice constant a0 of NaCl and LiF. Principles Bragg’s law of reflection describes the diffraction of plane waves at a monocrystal as the selective reflection of the waves at a set of lattice planes within the crystal. Due to the periodicity of the crystal, the lattice planes of a set have a fixed spacing d. An incident wave with the wavelength l is reflected with maximum intensity when the Bragg condition n ⋅ l = 2 ⋅ d ⋅ sinq (I) n: diffraction order l: wavelength d: spacing of lattice planes is fulfilled (see experiment P6.3.3.1). The angle q shows the direction of the incident and reflected wave with respect to the set of lattice planes and is often referred to as the glancing angle. In a cubic crystal with NaCl structure (cf. Fig. 1), the lattice planes run parallel to the surfaces of the crystal’s unit cells in the simplest case. Their spacing d corresponds to one half the lattice constant: d = a0 2 (II) This lets us use (I) as an equation for determining the lattice constant a0: n ⋅ l = a0 ⋅ sinq (III) In other words, to determine a0 we need to measure the glancing angle q for a known wavelength l and diffraction order n. This method is more precise when the glancing angles are also measured in higher diffraction orders. In this experiment, the molybdenum x-rays are used as radia￾tion of a known wavelength. Table 1 shows its wavelengths l. A Geiger-Müller counter tube is used to detect the x-rays; this instrument and the crystal are both pivoted with respect to the incident x-ray beam in 2q coupling – the counter tube is turned by twice the angle of the crystal (cf. Fig. 2). The zero point q = 08 is characterized by the fact that the lattice planes and the Table 1: Wavelengths of the characteristic x-ray radiation of molybdenum (weighted means [1]) Line l pm Ka 71.08 Kb 63.09 P7.1.2.1 1108-Ste 1