Landau parameter of elasticity

Based on the consideration given by the Ginzburg-Landau (GL) theory according to the variational principle, we assume that the microscopic Gibbs function density given by [1] ∫VGsdV = ∫(Fs - 1/4pBH)dv must be stationary at the thermodynamical equilibrium. To describe the universal propagation of th...

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Datum:	2006
Hauptverfasser:	Merabtine, N., Bousnane, Z., Benslama, M., Boussaad, F.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2006
Schriftenreihe:	Semiconductor Physics Quantum Electronics & Optoelectronics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/121610
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Zitieren:	Landau parameter of elasticity / N. Merabtine, Z. Bousnane, M. Benslama, F. Boussaad // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 3. — С. 1-3. — Бібліогр.: 4 назв. — англ.

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spelling	irk-123456789-1216102017-06-16T03:03:55Z Landau parameter of elasticity Merabtine, N. Bousnane, Z. Benslama, M. Boussaad, F. Based on the consideration given by the Ginzburg-Landau (GL) theory according to the variational principle, we assume that the microscopic Gibbs function density given by [1] ∫VGsdV = ∫(Fs - 1/4pBH)dv must be stationary at the thermodynamical equilibrium. To describe the universal propagation of the order parameter, we express order phases and amplitudes as dealing with tensor elements. In addition to the variation of the order parameter and the vector potential limited by the condition )()( xBxArrr =×∇ , we introduce here the concept of elasticity to describe the propagation of the superconducting state as “the little waves borning on smooth Superconductor Sea [2]”. The coherence concept transits to the asymptotic behaviour, we shall say that equivalence concept is its limit, this must transgress the propagation laws of superconductivity to be replaced by the increasing of superconductivity. Superconductivity will be viewed as second order extensive value, propagation seems to be so quick to avoid the stability, the increasing of superconductivity requires more time, and more time will be equivalent to a second and added measurement process eliminating the degeneracy of the first integral during the cooling process. It may deal with the first approximated stability of Superconductor State. The uncertainly in quantum mechanics is limited as scale length relations for the dimension coherence of the order parameter and temperatures. 2006 Article Landau parameter of elasticity / N. Merabtine, Z. Bousnane, M. Benslama, F. Boussaad // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 3. — С. 1-3. — Бібліогр.: 4 назв. — англ. 1560-8034 PACS 74.20.-z http://dspace.nbuv.gov.ua/handle/123456789/121610 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Based on the consideration given by the Ginzburg-Landau (GL) theory according to the variational principle, we assume that the microscopic Gibbs function density given by [1] ∫VGsdV = ∫(Fs - 1/4pBH)dv must be stationary at the thermodynamical equilibrium. To describe the universal propagation of the order parameter, we express order phases and amplitudes as dealing with tensor elements. In addition to the variation of the order parameter and the vector potential limited by the condition )()( xBxArrr =×∇ , we introduce here the concept of elasticity to describe the propagation of the superconducting state as “the little waves borning on smooth Superconductor Sea [2]”. The coherence concept transits to the asymptotic behaviour, we shall say that equivalence concept is its limit, this must transgress the propagation laws of superconductivity to be replaced by the increasing of superconductivity. Superconductivity will be viewed as second order extensive value, propagation seems to be so quick to avoid the stability, the increasing of superconductivity requires more time, and more time will be equivalent to a second and added measurement process eliminating the degeneracy of the first integral during the cooling process. It may deal with the first approximated stability of Superconductor State. The uncertainly in quantum mechanics is limited as scale length relations for the dimension coherence of the order parameter and temperatures.
format	Article
author	Merabtine, N. Bousnane, Z. Benslama, M. Boussaad, F.
spellingShingle	Merabtine, N. Bousnane, Z. Benslama, M. Boussaad, F. Landau parameter of elasticity Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Merabtine, N. Bousnane, Z. Benslama, M. Boussaad, F.
author_sort	Merabtine, N.
title	Landau parameter of elasticity
title_short	Landau parameter of elasticity
title_full	Landau parameter of elasticity
title_fullStr	Landau parameter of elasticity
title_full_unstemmed	Landau parameter of elasticity
title_sort	landau parameter of elasticity
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/121610
citation_txt	Landau parameter of elasticity / N. Merabtine, Z. Bousnane, M. Benslama, F. Boussaad // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 3. — С. 1-3. — Бібліогр.: 4 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT merabtinen landauparameterofelasticity AT bousnanez landauparameterofelasticity AT benslamam landauparameterofelasticity AT boussaadf landauparameterofelasticity
first_indexed	2025-07-08T20:13:05Z
last_indexed	2025-07-08T20:13:05Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 3. P. 1-3. © 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 1 PACS 74.20.-z Landau parameter of elasticity N. Merabtine1, Z. Bousnane2, M. Benslama1, F. Boussaad2 1Electromagnetism and Telecommunication Laboratory, Electronics Department, Faculty of Engineering, University of Constantine, 25000, Algeria 2Physics Department, Faculty of Science, University of Batna, 05000, Algeria E-mail: na_merabtine@hotmail.com; malekbenslama@hotmail.com Abstract. Based on the consideration given by the Ginzburg-Landau (GL) theory according to the variational principle, we assume that the microscopic Gibbs function density given by [1] dVHBFdVG V S V S ∫∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅−= rr π4 1 must be stationary at the thermodynamical equilibrium. To describe the universal propagation of the order parameter, we express order phases and amplitudes as dealing with tensor elements. In addition to the variation of the order parameter and the vector potential limited by the condition )()( xBxA rrr =×∇ , we introduce here the concept of elasticity to describe the propagation of the superconducting state as “the little waves borning on smooth Superconductor Sea [2]”. The coherence concept transits to the asymptotic behaviour, we shall say that equivalence concept is its limit, this must transgress the propagation laws of superconductivity to be replaced by the increasing of superconductivity. Superconductivity will be viewed as second order extensive value, propagation seems to be so quick to avoid the stability, the increasing of superconductivity requires more time, and more time will be equivalent to a second and added measurement process eliminating the degeneracy of the first integral during the cooling process. It may deal with the first approximated stability of Superconductor State. The uncertainly in quantum mechanics is limited as scale length relations for the dimension coherence of the order parameter and temperatures. Keywords: superconductivity, order parameter, elasticity. Manuscript received 21.02.06; accepted for publication 23.10.06. 1. Introduction We introduce relations between order parameter features and elasticity qualities. Elasticity expression of the second order, as an effective language translated to make a new equivalence between quantum mechanics descrip- tion and thermodynamical one which unfortunately is severely limited by the fundamental requirements of reversibility of processes and irreversibility of time. This will be considered as a very little step to express the asymptotic coherence of classical equations. 2. Order phases and amplitude partial derivative equations The superconducting state, viewed as perfect mixing wave functions regulated by a distribution of potentials, seems to be a result depending on temperature. As giving by classical equations, the free energy density and the order parameter expressed by equality meaning equivalence. These limitations imposed to this equivalence by the requirements presented by the ratio between the lengthscales of temperatures, just near the transition points. This fact was mentioned by Kenneth Wilson in 1972. The temperature acts as a deformation causing a phase transition. The order parameter given by [3] )()(η)( rierr φ=Ψ (1) is replaced by )()(η)( ki rri kiki errrr −−=−Ψ φ . (2) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 3. P. 1-3. © 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 2 The adjacent in levels distance is given by [3], )()( ESeEED −⋅Δ= (3) and )()( ESeEED ′−⋅Δ=′ . (4) E and E′ are two centres of two intervals of the energy. The distance D is universal for macroscopic bodies. The elasticity means the existence of D(F) given by D(F) = D(E) – D(TS(E)) , D(F) = [ ]TkES eTSSTEe B)()( −− Δ+Δ−Δ . (5) The expression TkeTSSTE B)( −Δ+Δ−Δ must be written as ΔΩ , where Ω is a function of S =ΔΩ TkeTSSTE B)( −Δ+Δ−Δ , =ΔΩ TkeTSQE B)( −Δ+Δ−Δ . (6) ΔΩ acts as a potential for cTT = , revealing the power of statistical weights. The electrons having a mean role obey to )( ln TSQ E Δ+ΔΔΩ Δ ~ kBTc . (7) The expression )()( ETSeTSST −Δ+Δ− is equivalent to 2uikdridrk. uik is an effective deformation tensor measuring the macroscopicity scale of the body, its form is as ki ki rr rr ∂∂ −Ψ∂ )(2 . We have: )()( ETSeTSST −Δ+Δ− ~ )(rie φ . )( )( )( 22 ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ∂∂ −φ∂ η+ ∂∂ −η∂ ki ki ki ki rr rr r rr rr (8a) When ( ) )(222 ESTr =φ , (8b) )( TSST Δ+Δ− = ki ki ki ki rr rr r rr rr ∂∂ −∂ + ∂∂ −∂ )( )(η )(η 22 φ . (9) Eq. (8b) expresses the equivalence between the phase and entropy. Eq. (8a) expresses the partial derivatives combination of the phase and amplitude, as generated by the thermodynamical quantity .)( TSST Δ+Δ− We write (8a) )()( ETSeTSST −Δ+Δ− ~ 2 )(rie φ , )( )( )( 22 ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂∂ −∂ + ∂∂ −∂ ki ki ki ki rr rr r rr rr φ η η (10) also )(1 ETSe T S S T − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ Δ + with ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ Δ T S S T <<1 that expresses the limit of the second member of Eq. (10). The use of such consideration is an essay to describe the features of the order parameter as given by ( ) 2rΨ , meaning the phase density as a result of the ordinary phase submitted to the potential Ω , we can consider it as a distribution of phases, or to be called the phase order. 3. Constraint tensor Having the variation principle that limits the causes of appearance of the order parameter through the whole sample, dVHBFdVG V S V S ∫∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ π −= rr 4 1 . We introduce the following notation, HBFS rr ⋅− π4 1 ~ ik ik ik i rr σ ∂ ∂ ∂ ∂ σ . We write dV rr dVG V k ik i ik V S ∫∫ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ σ∂ ∂ σ∂ = = Oσ2 dA A ik∫ . (11) A is an extremal area supporting the maximum order constraints. The momentum of forces causing the transition over the whole sample is given by dVr r r r M V i k ik k i ik ik ∫ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ = σσ . (12) dV is an extremal volume where these order constraints are observed. The fundamental thermodynamical relation can be written as follows [4]: dE = TdS + ikik duσ . For a weak deformation, the iku tensor is a linear function of the constraint tensor ikσ , which is the Hooke law for the elasticity of the density free energy. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 3. P. 1-3. © 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 3 The general law [4] is ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ σδ−σ μ +σδ= 1111 3 1 2 1 9 1 ikikikik K u . Where K is a compression amplitude, and μ is the Lame constant. ikσ is the solution of the equation )(rie φ = ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ∂∂ −φ∂ η+ ∂∂ −η∂ ki ki ki ki rr rr r rr rr )( )( )( 22 .σδ 3 1 μ2 1σδ 9 1 1111 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+= ikikikK σ (13) ikσ will be function of η and φ . K and μ are functions of Landau phenomenological constants a and b. 4. Thermodynamical function expressed by tensors .σ ikik duSdTdF +−= The thermodynamical potential TSE −=Φ = ikσ− )(rie φ × ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ∂∂ −∂ + ∂∂ −∂ ki ki ki ki rr rr r rr rr )( )(η )(η 22 φ . The expression of Φ means the creation or annihilation of levels, or a set of levels. This fact is interpreted as the interaction of scale lengths, when the phase order is regulated by the Dirac function. In such way )(δ\|)(\| 0 2 rrdrr −=∫ φ , 0r is the mean radius over which the superconductor state propagates. ( ) 1δ 0 =− rr , the appearance of a set of levels containing an extremum number of levels. ( ) 0δ 0 =− rr , the disappearance of a set containing the extremum number of levels. (Extremum means a minimum number.) The use of such consideration concerning the effective elasticity of the order parameter leads to draw the propagation of this parameter as “peaceful sea with little waves against the beach”. Those little waves characterised by elasticity of the amplitude and phase, seems to be so weak to preserve the stability of Superconductor State when temperature increases. 5. Interpretation of the phase order equivalence equation It determines “the levels of macroscopicity of the sample”. The coherence scale between two macroscopic levels is as 2T . The permitted levels are given by ∫ = .min)(2 dEEST The macroscopic levels are different ways to express the recombination of entropies, every time the macroscopic level fluctuates. The partial derivative equation amplitude, makes a meaning of an effective free mean path of the second order, in a polar manifold which corresponds to the maximum interval EΔ , where the elasticity of free energy density is undeterministically collected as to give locally iku , having the components excluding each other (we can reduce this interval EΔ of maximum number of levels, to a minimum number of levels under iku tensor, the ikσ tensor will cause the reverse production of levels to reach the initial maximum number of levels). 6. Conclusion The concept of elasticity introduced above is a little tendency to express the non-absolutism of propagation of the order parameter and show that the universality of undetermination considered in quantum mechanics for macroscopic bodies must be reformulated as a limitation imposed to the scale coherence by Landau and temperature, in such a way to evaluate a constant governing the precision under which the scale coherence and the temperature must be measured. References 1. M. Thinkham, Introduction to superconductivity. Second Edition. McGraw Hill Inc, 1996. 2. Charles W. Misner, Kip S. Thorne and J.A. Whee- ler, Inspired from vacuum fluctuations. “Gravitation” Company, San Fransisco, 1973. 3. L. Landau, E. Lifshitz, Statistical Physics. Edition Mir, Moscow, 1967. 4. L. Landau, E. Lifshitz, Theory of elasticity. Edition Mir, Moscow, 1989.

Landau parameter of elasticity

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