The Crystal Lattice: Phonons, Solitons, Dislocations, by Arnold M. Kosevich
By Arnold M. Kosevich
The purpose of this winning publication 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 specializes in low-dimensional types of crystals and on superlattices. either conventional questions just like the spectrum of vibrations, the belief of phonon fuel, dislocations and so forth. and new elements just like the concept of quantum crystals, solitons in 1D crystals, dislocation idea of melting of second crystals and so on. are mentioned. the writer provides a proof of a suite of phenomena which entered into stable nation physics over the last many years. it truly is proven that the crystal houses are delicate to the measurement of the crystal and its disorder constitution, and count a bit of on no matter if the periodic constitution comprises atoms, or electric dipoles, or magnetic moments (spins). huge consciousness is dedicated to the dislocation mechanisms as a foundation of the idea of plasticity and diverse technological purposes of crystalline fabrics.
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Extra info for The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices
14) whose solution is any double-differentiated function depending on the argument x − st (if the wave runs in the positive direction of the axis x) or x + st (if the wave runs in the opposite direction). We remind readers that s is the sound velocity. But if the harmonic approximation is insufﬁcient, the answer to the question posed is no longer obvious. 12). Since a transfer of any deformation impulse in a 1D crystal is connected with motion of a local compression, the analysis of dynamics of the derivative p = u x makes sense supposing that the plot of p(x) has a form similar to Fig.
4) The equations given admit one trivial integration. 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.
13). 13) that contains ∂u/∂x is also small. 13) in a linearized form is reduced to vtt = 0 svxx − a2 A2 vxxxx. 15) is associated with the dispersion relation ω2 = 0s 2 2 k + a2 A2 k4 . 13) but with the opposite sign in front of the term proportional to k4 . 17) 0 (sk) = s∗ k. √ Its sound velocity s∗ = s 0 is small. Thus, the long-wave dispersion law of bending vibrations of a linear chain that experienced a static longitudinal stretching does not differ qualitatively from the dispersion law of longitudinal vibrations of this chain.