By Arnold M. Kosevich(auth.)
The purpose of this winning e-book is to explain and examine peculiarities of classical and quantum dynamics of a crystal as a spatially periodic constitution. within the moment revised and up-to-date version, the writer makes a speciality of low-dimensional versions of crystals and on superlattices. either conventional questions just like the spectrum of vibrations, the assumption of phonon gasoline, dislocations and so forth. and new facets just like the concept of quantum crystals, solitons in 1D crystals, dislocation concept of melting of second crystals and so on. are mentioned. the writer provides an evidence of a suite of phenomena which entered into strong kingdom physics over the last many years. it truly is proven that the crystal homes are delicate to the measurement of the crystal and its disorder constitution, and rely a bit on even if the periodic constitution comprises atoms, or electric dipoles, or magnetic moments (spins). massive cognizance is dedicated to the dislocation mechanisms as a foundation of the speculation of plasticity and various technological functions of crystalline materials.Content:
Chapter 1 Mechanics of a One?Dimensional Crystal (pages 15–57):
Chapter 2 basic research of Vibrations of Monatomic Lattices (pages 59–88):
Chapter three Vibrations of Polyatomic Lattices (pages 89–123):
Chapter four Frequency Spectrum and Its reference to the fairway functionality (pages 125–151):
Chapter five Acoustics of Elastic Superlattices: Phonon Crystals (pages 153–162):
Chapter 6 Quantization of Crystal Vibrations (pages 163–181):
Chapter 7 interplay of Excitations in a Crystal (pages 183–201):
Chapter eight Quantum Crystals (pages 203–212):
Chapter nine element Defects (pages 213–232):
Chapter 10 Linear Crystal Defects (pages 233–246):
Chapter eleven Localization of Vibrations (pages 247–277):
Chapter 12 Localization of Vibrations close to prolonged Defects (pages 279–296):
Chapter thirteen Elastic box of Dislocations in a Crystal (pages 297–319):
Chapter 14 Dislocation Dynamics (pages 321–340):
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Extra info for The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition
The last observation will be supported with a certain qualitative argument of general character. It follows from a comparison of the properties of solitons of the two types considered by us. Irrespective of the character of nonlinearity that generates a soliton, the value of its limiting velocity is completely determined by the dispersion law of harmonic vibrations that can exist in the system under study. 13), it is characterized by some maximum phase velocity s0 . The velocity of a soliton (if it arises) will exceed s.
To perform this, we assume the linear chain experiences a static longitudinal stretching (longitudinal strain) equal to ε 0 (ε 0 1). Besides that, as we are interested primarily in the solitary waves, we assume all velocities and all gradients vanish at inﬁnity. 5) ; ε 0 = constant. 4) that (V 2 − ε 0 )vx + aA s 2 vxxx = V 2 vx2 vx . 6) involves only the derivatives of the vector function v and denote w = vx . 7) where V2 1 s s V2 − ε0 . , γ= 2 2 aA 1 − V aA We introduce the amplitude and the phase ϕ of the transverse motion velocity by means of the relation w = w(i1 cos ϕ + i2 sin ϕ), β= where i1 , i2 are the unit vectors of two coordinate axes perpendicular to the direction of a nondeformed chain.
Solution. 23) where α1 and α2 are parameters of the elastic interaction between the nearest and nextnearest neighbor atoms, respectively. 18) becomes similar to Fig. 2. 2. Find the wave number values for the frequencies exceeding the maximum frequency of harmonic 1D crystal vibrations with interaction of nearest neighbors only. Interpret the result. Hint. 4) ω is real, ﬁnd the complex k values corresponding to ω > ωm . Solution. 24) where κ is determined by aκ . 25) 2 The solutions exponentially decreasing (or increasing) with the distance can describe the vibrations of a bounded 1D crystal that are localized near its free edge and not penetrating inside.