Neutron Scattering - JUWEL - Forschungszentrum Jülich

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POWDER DIFFRACTOMETER SPODI 5 T ( Q) j � 1 exp�� 2 � � �� Qu � j vector uj reflects the thermal displacements of atom j Braggs' Law Braggs' Law provides a relation between distances of lattice planes with Miller indexes hkl, i.e. dhkl, and the scattering angle 2� of the corresponding Bragg peak. Braggs' law can be illustrated in a simplified picture of diffraction as reflection of neutron waves at lattice planes (figure 4). The waves which are reflected from different lattice planes do interfere. We get constructive interference, if the path difference between the reflected waves corresponds to an integer multiple of the wavelength. The condition for constructive interference (= Braggs' law) is then: 2 d hkl sin� � n� Figure 1: Illustration of Bragg’s law: constructive interference of neutron waves, reflected from lattice planes, where �, 2� are Bragg angles, 2�=2dhklsin� is the path difference and 2�=n� is the constructive interference. Applying Bragg’s law one can derive the lattice spacings (“d-values“) from the scattering angle positions of the Bragg peaks in a constant-wavelength diffraction experiment. With the help of d-values a qualitative phase analysis can be carried out.
6 POWDER DIFFRACTOMETER SPODI Ewald's sphere The Ewald's sphere provides a visualisation of diffraction with help of the reciprocal lattice. At first, we introduce the scattering vector Q and the scattering triangle (Figure 2). The incident neutron wave is described by a propagation vector ki, the scattered wave is given by kf. In the case of elastic scattering (no energy transfer) both vectors ki and kf have the same length which is reciprocal to the wavelength. k i k � f 2� � � remark: The length of the wave vectors are sometimes given as k i � k f 1 � (This definition is found � esp. in crystallographic literature, while the other one is more common for physists). The angle between vectors ki and kf is the scattering angle 2�. The scattering vector Q is the given by the difference between ki and kf : Q= k f � k i � � � sin Q � 4 Figure 2: Illustration of scattering vector and scattering angle resulting from incident and scattered waves. In the visualisation of the diffraction phenomena by Ewald the scattering triangle is implemented into the reciprocal lattice of the sample crystal – at first, we consider diffraction at a single crystal (Figure 3). Note that the end of the incident wave vector coincides with the origin of the reciprocal lattice. Ewald revealed the following condition for diffraction: we have diffraction in the direction of kf, if its end point (equivalently: the end point of scattering vector Q) lies at a reciprocal lattice point hkl. All possible kf, which fulfil this condition, describe a sphere with radius ��, the so called Ewald's sphere. Thus we obtain a hkl reflection if the reciprocal lattice point hkl is on the surface of the Ewald's sphere.
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SPHERES Backscattering spectrometer
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SPHERES 3 1 Introduction Neutron ba
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SPHERES 5 Fig. 2: The monochromator
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SPHERES 7 While the primary spectro
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SPHERES 9 neutron is scattered, it
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SPHERES 11 The stationary equilibri
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SPHERES 13 4. Summarize the tempera
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________________________ DNS Neutro
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DNS 3 1 Introduction Polarized neut
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DNS 5 Monochromator horizontal- and
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DNS 7 3 Preparatory Exercises The p
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DNS 9 important and always necessar
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DNS 11 Contact �� Phone:
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________________________ KWS-3 Very
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KWS-3 3 1 Introduction Ultra small
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KWS-3 5 2. The standard Q-range of
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KWS-3 7 Table 2. Samples for practi
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KWS-3 9 References [1] Eskilden, M.
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________________________ RESEDA Res
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RESEDA 3 1 Introduction Neutrons ar
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RESEDA 5 various length values l ac
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RESEDA 7 In this expression, a norm
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RESEDA 9 spin flip from up to down
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RESEDA 11 The neutrons are counted
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RESEDA 13 Contact ��
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