^{1}

^{*}

^{2}

^{3}

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.

With the advent of quantization of the motion of dislocations due to lattice vibrations by Mott [

where the frequency is

constant,

thermally activated creep. They approximated

is the mass of an electron and b is the Burgers vector. Their original formulation for quantizing the motion of dislocations with lattice vibrations contradicts the approximation for mass of the electron. They would have considered the quanta of lattice vibrations by considering the mass of the phonon. The mass of the phonon can be determined by knowing the frequency of phonon under damped conditions (Bardoni resonant peaks) for depinning of dislocations from Peierls barrier. Using

where m will be considered for the mass of phonon instead of electron and b is the Bourger’s vector. where

The quantum mechanical effects in a single barrier stochastic model were allowed for stress relaxation [

where

where

enough for a limited low temperature range, i.e.,

Using the simple or single barrier stochastic model of logarithmic creep of Buckle and Feltham [

The quantized Peirels barrier width or quantum length [

where in Equations (6)-(8),

The anomalies in the temperature dependance of yield stress of metals at low temperature were studied [

Usually such anomalies are experimentally reported in the temperature range of

Debye temperature (follows quantum theory of the specific heat of metals) in kelvin. Raza [

where

A sudden escalation in the yield stress below about 10 K, following a linear profile of stress relaxation and indeed of yield stress with a negative slope, can be ascribed to quantum elasticity as a manifestation of stress causing effects, i.e.,

Now, we modify such observations [

is considered as the mass of phonon at different Debye temperature ranges.

Using

Equations (9)-(11) are self explanatory to reflect quantum mechanical yielding of crystals at different Debye temperatures.The quantum elasticity is confirmed from Equation (11) where pinned dislocations from Peierls barrier are stretched with a “quantum action,

the Gaussian region, i.e.,

and directly proportional to

The logarithmic creep which Weertmann [

Taking natural logarithm of Equation (2), we have

The last two terms in Equation (12) are explicit and have nothing to do with creep rate, therefore, they are neglected. Hence, we have

Putting Equation (13) in Equation (4), we have

Equation (14) shows that the logarithmic creep rate is interpreted as stress relaxation rate ( stress relaxation curves in crystal plasticity). Equation (14) provides the slope, i.e., “s” of relaxation curve. Surprising all the relaxations curves when plotted and checked on semi-log or log-log graph papers shows the logarithmic profile which is a confirmation to the validity of the result of Mott [

Equation (14), i.e.,

shows fluence, i.e., the number of depinned dislocations crossing the area per unit time whereas the term “

With Equation (8) of quantized Peierls barrier width, i.e.,

of time, i.e.,

will provide the rate determining processes of dislocations (some of them quantized on Peierls barriers, whereas

others are dipinned from Peierls barriers) in the curvature of space

activation energy values for both of the process at the quantum level. Remember that the curvatures of space

where

where

Theories of Mott and Weertmann pertaining to quantum mechanical tunneling of dislocations from Peierls barrier in cubic crystal are reconsidered. The quantum mechanical yielding in metals at relatively low temperature is resolved and formula is presented. The crystal plasticity is studied in terms of stress relaxation curves and the formula is presented. Formulas for simultaneous quantization of dislocations with their stress and depinning of dislocations are presented.

Saleem Iqbal,Farhana Sarwar,Syed Mohsin Raza, (2016) Quantum Mechanical Tunneling of Dislocations: Quantization and Depinning from Peierls Barrier. World Journal of Condensed Matter Physics,06,103-108. doi: 10.4236/wjcmp.2016.62014