Multiple Diffraction of X-Rays in Crystals

Chang, Shih‐Lin

doi:10.1007/978-3-642-82166-0

Cited by 225 publications

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“…A stopgap is also formed when Bragg diffraction occurs on multiple Bragg planes simultaneously, which is called multiple-Bragg diffraction [11], and is fundamental for bandgap formation [2,12,13]. Wave propagation in crystals is described along high-symmetry directions [1].…”

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confidence: 99%

Observation of Sub-Bragg Diffraction of Waves in Crystals

et al. 2012

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We investigate the diffraction conditions and associated formation of stopgaps for waves in crystals with different Bravais lattices. We identify a prominent stopgap in high-symmetry directions that occurs at a frequency below the ubiquitous first-order Bragg condition. This sub-Bragg diffraction condition is demonstrated by reflectance spectroscopy on two-dimensional photonic crystals with a centred rectangular lattice, revealing prominent diffraction peaks for both the sub-Bragg and first-order Bragg condition. These results have implications for wave propagation in 2 of the 5 two-dimensional Bravais lattices and 7 out of 14 three-dimensional Bravais lattices, such as centred rectangular, triangular, hexagonal and body-centred cubic.The propagation and scattering of waves such as light, phonons and electrons are strongly affected by the periodicity of the surrounding structure [1,2]. Frequency gaps called stopgaps, emerge for which waves cannot propagate inside crystals due to Bragg diffraction. Bragg diffraction is important for crystallography using X-ray diffraction [3] and neutron scattering [4]. Diffraction determines electronic conduction of semiconductors [1,2] and of graphene [5], and broad gaps are fundamental for acoustic properties of phononic crystals [6,7] and optical properties of photonic metamaterials [9, 10].Bragg diffraction is described in reciprocal space by the Von Laue condition − → Here m is an integer, λ is the wavelength inside the crystal, θ is the angle of incidence with the normal to the lattice planes, and d is the spacing between the lattice planes. A stopgap is also formed when Bragg diffraction occurs on multiple Bragg planes simultaneously, which is called multiple-Bragg diffraction [11], and is fundamental for bandgap formation [2,12,13]. Wave propagation in crystals is described along high-symmetry directions [1]. Multiple-Bragg diffraction has been recognized in highsymmetry directions at frequencies above the first-order simple Bragg diffraction condition: m = 1, λ = 2d orhas not yet been observed at frequencies below simple Bragg diffraction [14].In this Letter we show that for high-symmetry directions multiple-Bragg diffraction can occur at frequencies below the first order simple Bragg condition. As a demonstration we have investigated diffraction conditions for two-dimensional (2D) photonic crystals using reflectance spectroscopy. A broad stopgap is observed below the simple Bragg condition, depending on the symmetry of the lattice. Our findings are not limited to light propagation, but apply for wave propagation in general, and therefore we anticipate similar diffraction for electrons in graphene [5], and sound in phononic crystals [6,7].We have studied light propagation in 2D silicon photonic crystals [17]. Figure 1(a) shows a scanning electron microscope (SEM) image of one of these crystals from the top view. The centred rectangular unit cell has a long side a = 693 ± 10 nm and a short side c = 488 ± 11 nm. The pores have a radius of r = 155 ± 10 nm and are approxima...

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confidence: 99%

Observation of Sub-Bragg Diffraction of Waves in Crystals

et al. 2012

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“…The calculation procedure given by Chang (1984) is employed. Since the calculation of the present case involves the dispersion surface in reciprocal space and the direction of Poynting vectors in real space, the coordinates chosen for the reciprocal and the real space are shown in Figs.…”

Section: Calculationsmentioning

confidence: 99%

“…These include the work of Saccocio & Zajac (1965a, b), Hildebrandt (1967), Joko & Fukuhara (1967), , , Uebach & Hildebrandt (1969), Balter, Feldman & Post (1971), Umeno & Hildebrandt (1975), Post, Chang & Huang (1977), H¢ier & Aanestad (1981), and many others. Fairly complete references are given by Chang (1984). Most of these deal with anomalous transmission in thick crystals.…”

Section: Introductionmentioning

confidence: 99%

Experimental observation of the Borrmann pyramid – the Borrmann fan for four-beam transmission of X-rays

Campos¹,

Chang²

1986

Acta Cryst Sect A

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348COMPARISON OF UMWEGANREGUNG AND CALCULATED PATTERNS radiation. The reciprocal-lattice points 323 and 320 do not pass the Ewald sphere belonging to the wavelength of Cu Ka2, but when the reciprocallattice points are replaced by spheres, these spheres touch the Ewald sphere of Cu Ka2 and contribute to the intensity in the above-mentioned region. Comparison of measured and calculatedUmweganregung patterns Fig. 2(b) is calculated for Cu Kal radiation, Fig. 2(c) for Cu Kot 2 radiation, qJ is zero in the [100] direction. With these two diagrams it is possible to index all Umweganregung peaks of the measured pattern in Fig. 2(a). The indices of the operative and cooperative reflections are given in Tables 1 and 2 together with their Bragg angles and structure factors. The zero point for qJ in the measured diagram is chosen arbitrarily. The leftmost peak in Fig.2(a) is built up of the two operative reflections 221 and 27-4 with Cu Kal radiation. The second peak belongs to the same reflections, but for Cu Ka2 radiation. The difference in qJ for these two peaks is qJ~,-qJ-2 =-0'42°, as can be calculated from values in Tables 1 and 2. The third peak is the peak discussed above, having a respective triangle in Fig. 2(b) only.Despite the difficulty of correcting for the superposition of the Cu Kal and Cu Ka2 peaks and disregarding the problematical second peak in Table 1, one finds that the comparison of the measured and calculated intensities gives surprisingly good agreement. The disagreement between the measured and calculated intensities for the peak at 0 = 52.18 ° is probably due to experimental shortcomings. As can be seen in Fig. 1

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“…the term D (n) in the left-hand side (3) corresponds to n-wave Bragg dynamical diffraction [4] (equation D (n) = 0 is the dispersion equation defining diffraction modes in n-wave case).…”

Section: Introductionmentioning

confidence: 99%

Use of dynamical undulator mechanism to produce short wavelength radiation in volume FEL (VFEL)

Baryshevsky

Батраков²

2003

Nuclear Instruments and Methods in Physics Research Section A:

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VFEL lasing in system with dynamical undulator is described. In this system radiation with longer wavelength creates the undulator for lasing on shorter wavelength.Two diffraction gratings with different spatial periods form VFEL resonator. The grating with longer period pumps the resonator by radiation of longer wavelength to provide necessary amplitude of undulator field. The grating with shorter period makes mode selection for radiation of shorter wavelength. Lasing of such a system in terahertz frequency range is discussed.

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Multiple Diffraction of X-Rays in Crystals

Cited by 225 publications

References 0 publications

Observation of Sub-Bragg Diffraction of Waves in Crystals

Observation of Sub-Bragg Diffraction of Waves in Crystals

Experimental observation of the Borrmann pyramid – the Borrmann fan for four-beam transmission of X-rays

Use of dynamical undulator mechanism to produce short wavelength radiation in volume FEL (VFEL)

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