Abstract
Theories of Mott and Weertmann pertaining to quantum mechanical tunneling of dislocations from Peierls barrier in cubic crystals are revisited. Their mathematical calculations about logarithmic creep rate and lattice vibrations as a manifestation of Debye temperature for quantized thermal energy are found correct but they can not ascertain to choose the mass of phonon or “quanta” of lattice vibrations. The quantum mechanical yielding in metals at relatively low temperatures, where Debye temperatures operate, is resolved and the mathematical formulas are presented. The crystal plasticity is studied with stress relaxation curves instead of logarithmic creep rate. With creep rate formulas of Mott and Weertmann, a new formula based on logarithmic profile of stress relaxation curves is proposed which suggests simultaneous quantization of dislocations with their stress, i.e., and depinning of dislocations, i.e., , where is quantum action, σ is the stress, N is the number of dislocations, A is the area and t is the time. The two different interpretations of “quantum length of Peierls barrier”, one based on curvature of space, i.e., yields quantization of Burgers vector and the other based on the curvature of time, i.e., yields depinning of dislocations from Peierls barrier in cubic crystals, are presented. , i.e., the unitary operator on shear modulus yields the variations in the curvature of time due to which simultaneous quantization, and depinning of dislocations occur from Peierls barrier in cubic crystals.
Highlights
With the advent of quantization of the motion of dislocations due to lattice vibrations by Mott [1], a new idea following the same treatment to the case of dislocations crossing Peierls barrier was floated by Weertmann [2]
Quantum Mechanical Yielding in Metals at Relatively Low Temperatures
We investigated quantum mechanical tunneling of dislocations by considering results of Mott [1], Weertmann [2], Jafri et al [8], Majeed [9], Majeed et al [10] and Raza [3]-[5] [11] [12]
Summary
With the advent of quantization of the motion of dislocations due to lattice vibrations by Mott [1], a new idea following the same treatment to the case of dislocations crossing Peierls barrier was floated by Weertmann [2]. Using the simple or single barrier stochastic model of logarithmic creep of Buckle and Feltham [7], the quantum behaviour and stochasticity of crystal plasticity for Peierls barrier in cubic metals were studied by Jafri et al [8] Their assumption that the Peierls barrier width during creep, preferably the logarithmic creep, at relatively low temperatures, i.e., 1.7 K ≤ T ≤ 300 K , would remain unchanged. Where in Equations (6)-(8), ν ′ is the frequency of phonon, ν is the Poissons ratio of crystal, b is the Burgers vector, μ is the shear modulus of cubic metals and a is the lattice of cubic crystals We shall apply these equations to develop new meaningful expressions for crystal yielding, plasticity or work hardening with stress relaxations curves, quantization of dislocations over Peierls barrier and depinning of dislocations from Peierls barrier in the form of logarithm flux per unit time, i.e., logarithms fluence
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