Neutron Scattering - JUWEL - Forschungszentrum Jülich

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PUMA 3 1. Introduction Excitations in crystals can be described using formalism of dispersion relations of the normal modes or quasi-particles (phonons, magnons, etc.). These relations contain the most detailed information on the intermolecular interactions in solids. The result of a neutron scattering experiment is the distribution of neutrons that have undergone an energy exchange �� = Ei - Ef, and a wave vector transfer, Q = ki – kf , after scattering by the sample.: d � � ( 2 , ) � � ( , ) � ( , ) � d�d � 4 4 � 2 � k f � coh � inc � � N Scoh Q � Sinc Q � (1) � k � � i �coh is coherent scattering cross section, �inc is incoherent scattering cross section. They are constants that can be found in tables (http://www.ncnr.nist.gov/resources/n-lengths/). S(Q,�) functions depend only on the structure and dynamics of the sample and do not depend on the interaction between neutrons and the sample. Sinc(Q,�) reflects individual motions of atoms. Scoh(Q,�) provides the information on the structure and collective excitations in the sample. Energy transfer �� = Ei - Ef Momentum transfer k k Q � � The triple axis spectrometer is designed for measuring the Scoh(Q,�) in monocrystals. Therefore this function is of special interest for us. k � 0 2mE � f � 2� � 2 2 Q � k � k � 2k k i If ki = kf 4� Q � 2ki sin� � sin� � f i f cos 2�
4 O. Sobolev, A. Teichert, N. Jünke 2. Elastic scattering and Structure of Crystals In the case of coherent elastic scattering, when � = 0 (ki = kf ) only neutrons, that fulfill the Brags law are scattered by the sample: n� = 2dhklsin�hkl, (2) where � is a wavelength of neutron, dhkl is a distance between crystal planes described by corresponding Miller indexes hkl. �hkl denotes the angle between incoming (outgoing) scattering beam and the (hkl) plane. For the analysis of the scattering processes in crystals it is convenient to use the concept of the reciprocal space. For an infinite three dimensional lattice, defined by its primitive vectors a1, a2 and a3, its reciprocal lattice can be determined by generating three reciprocal primitive vectors, through the formulae: a2 � a3 g1 � 2� a � a � a g g 2 3 1 a1 � a3 � 2� a � a � a 2 a1 � a2 � 2� a � a � a 3 2 1 1 3 3 2 Note the denominator is the scalar triple product. Geometrically, the scalar triple product a1(a2�a3) is the volume of the parallelepiped defined by the three vectors. Let us imagine the lattice of points given by the vectors g1, g2 and g3 such that � is an arbitrary linear combination of these vectors: � � hg � kg � lg , (4) 1 2 3 where h,k,l are integers. Every point of the reciprocal lattice, characterized by � corresponds in the position space to the equidistant set of planes with Miller indices (h,k,l) perpendicular to the vector �. These planes are separated by the distance 2� d hkl � (6) � hkl The Brag’s condition for diffraction can be expressed in the following vector form: Q = �hkl (7) A useful construction for work with wave vectors in reciprocal space is the Brillouin zone (BZ). The BZ is the smallest unit in reciprocal space over which physical quantities such as phonon or electron dispersions repeat themselves. It is constructed by drawing vectors from one reciprocal lattice points to another and then constructing lines perpendicular to these vectors at the midpoints. The smallest enclosed volume is the BZ. (3)
Page 1 and 2: Member of the Helmholtz Association
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16 M. Meven 4.3 Oxygen Position The
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18 M. Meven To direct the secondary
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20 M. Meven For a given spectrum
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22 M. Meven References [1] Th. Hahn
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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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2 O. Holderer, M. Zamponi and M. Mo
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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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2 S. Mattauch and U. Rücker Conten
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4 S. Mattauch and U. Rücker Fig. 2
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6 S. Mattauch and U. Rücker 4 Expe
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8 S. Mattauch and U. Rücker Contac
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