Research Article Size Effects on Surface Elastic Waves in...

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2013 Article ID 528208 11 pageshttpdxdoiorg1011552013528208

Research ArticleSize Effects on Surface Elastic Waves in a Semi-Infinite Mediumwith Atomic Defect Generation

F Mirzade

Institute on Laser and Information Technologies Russian Academy of Sciences Moscow 140700 Russia

Correspondence should be addressed to F Mirzade fmirzaderamblerru

Received 3 May 2013 Accepted 10 October 2013

Academic Editor Michael C Tringides

Copyright copy 2013 F Mirzade This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper investigates small-scale effects on the Rayleigh-type surface wave propagation in an isotopic elastic half-space upon laserirradiation Based on Eringenrsquos theory of nonlocal continuum mechanics the basic equations of wave motion and laser-inducedatomic defect dynamics are derived Dispersion equation that governs the Rayleigh surface waves in the considered mediumis derived and analyzed Explicit expressions for phase velocity and attenuation (amplification) coefficients which characterizesurface waves are obtained It is shown that if the generation rate is above the critical value due to concentration-elastic instabilitynanometer sized ordered concentration-strain structures on the surface or volume of solids ariseThe spatial scale of these structuresis proportional to the characteristic length of defect-atom interaction and increases with the increase of the temperature of themedium The critical value of the pump parameter is directly proportional to recombination rate and inversely proportional todeformational potentials of defects

1 Introduction

During the last decades elastic wave propagation in solids(metals and semiconductors) with nonequilibrium atomicdefects has received a lot of attention [1ndash4] Various typesof lattice defects (vacancies and interstitials) produced fromthe lattice site atoms due to intense external energy fluxes(laser and corpuscular radiations) introduce a significantstrain of the medium as a result of the difference betweenthe radii of lattice atoms and defects [5] and play animportant role in surface modification of solids exposed tolaser radiation [1 5 6] The formation of atomic defects mayoccur also in a number of other technologies processes inthe laser fast recrystallization laser annealing multipulselaser etching pulsed laser-assisted thin-film deposition andso forth Strains in an elastic wave cause a strain-induceddrift of defects whereas the strains and a variation in thetemperature in the wavemodulate the rates of generation andrecombination of defects of the thermal-fluctuation origin(via variations in the energies of the defect formation andmigration) [5 6]

Several mathematical models have been considered tostudy the self-organization of two-dimensional (2D) ordered

microscale concentration-strain (CS) structures (at high con-centrations of atomic defects) on the surface of the solid half-space [7 8] and in an isotropic solid elastic layer [9 10] underthe action of laser irradiation In these studies attentionwas also focused on the study of an influence of the strain-induced diffusion generation and annihilation of defects onthe propagation characteristics of strain waves In both [1ndash4] and [7ndash10] the analysis was based on coupled evolutionequations for the atomic defect density fields and the classicallocal elasticity equations for the elastic displacement of themedium

Classical continuum theory is characterized by the localcharacter of stress (stress at a point depends only on the strainat that point) and does not contain internal length scale Theabsence of the length scale creates several discrepancies in thepredictions ofmechanical responses for example nondisper-sive wave behavior (constant wave velocity independent offrequency) For example according to the classical elasticityRayleigh waves propagating on the surface of a semi-infiniteisotropic elastic space are not dispersive at any frequencywhereas experiments and the atomic theory of lattice predictotherwise

2 Advances in Condensed Matter Physics

Limitations of the classical elasticity theory are alsodemonstrated in the study of the formation of coupled strain-defect nanometer sized ordered structures (short-range CSstructures) on the surface of laser-irradiated solids andmechanical behavior of microstructured materials becausetheir behavior is characterized by nonlocal stresses and theexistence of an internal length scale [11 12] The accurateanalysis of dynamic behaviors of these structures cannotbe correctly described by classical local elasticity theoryConducting experiments with nanoscale size materials isfound to be difficult and expensive Therefore develop-ment of appropriate nonlocal elasticity mathematical modelsfor nanostructure formation and nanomaterials is of greatimportance

In contrast to local approach of zero-range internalinteractions the nonlocal elasticity theory originated anddeveloped in the last four decades postulates that the stressat a point 119909 in a body depends not only on the strainat point 119909 but also on those at all other points of thebody Various nonlocal theories of linear elasticity have beenproposed to describe the scale effects on the characteristicsof the vibration and elastic wave propagation in the above-mentioned submicro- or nanosized structures The basicconstitutive equations and governing equations of linearnonlocal elasticity were derived by Eringen [11] and Kunin[12] Later the nonlocal theories have been applied for theanalysis of micro- and nanoscale plate-like structures inwhich the small-scale effects become significant In [13] Erin-gen considered vibration behavior of a nanoplate by usinglinear theory of nonlocal continuummechanics By using [11]Reddy [14] reformulated the classical and shear deformationbeam and plate theoriesThe nonlocal scale influences on thewave dispersion properties of the nanoplates are discussed indetail in [15] Vibration characteristics of nanoplates basedon three-dimensional theory of elasticity employing nonlocalcontinuummechanics have been discussed in [15] A reviewof some other applications of nonlocal elasticity theoriesfor nanostructures can be found in [16] Propagation oflongitudinal elastic and thermoelastic waves in an isotropichomogeneous infinite medium with long-rang interactionshas been studied by Nowinski [17 18] Other advances havebeen made by the application of nonlocal elasticity to suchfields as the solid defects [19 20] and fracture mechanics [21]

The present paper is concerned with the nonlocal elastic-ity theory of laser-excited solid half-space withmobile atomicpoint defects We summarise the theory formulated in [8] toanalyze the effects of nonlocal atom-atom and atom-defectinteractions on the surface wave propagation in solids withdefect generation The changes in defect concentration arecontrolled by the (i) generation of defects by laser irradia-tion (ii) their diffusion and (iii) strain-induced diffusionDispersion equation that governs the propagation of elastic-concentration waves has been derived by solving a system ofcoupled partial differential equations Some limiting cases ofthe dispersion equations are considered For a small value ofcoupled parameter obvious expressions for the phase velocityand the attenuation (amplification) factor of surface elastic-concentration waves are determined It is found that bothphase velocity and attenuation coefficient are to be influenced

by the presence of defect generation We also obtain thatat certain conditions concentration-elastic instabilities withthe formation of regular nanosized structures in a system ofatomic defects on the surface of the solids can be developedWe demonstrate that due to the nonlocal character of thedefect-atom and atom-atom interactions the dispersive curveof the instability has twomaxima As a result size distributionfunctions of the surface nanoscale nonhomogeneities havingtwomaxima can be formedThe results of some earlier worksare also deduced from the present formulation To our knowl-edge the problem in its present form was not investigatedbefore

2 Governing Equations

In this section we give basic 2D equations governing thedefect density and elastic displacement fields based on thenonlocal constitutive relations of the elasticity theory Con-sider an isotopic elastic semi-infinite medium occupying aregion minusinfin lt 119909 lt infin 0 lt 119911 lt infin in a rectangular Cartesiancoordinate system 119874 119909 119910 119911 where the origin 119874 is situated atany point on the plane boundary and 119874119911 points verticallydownwards that is towards the bulk of the medium Thesurface 119911 = 0 is supposed to be free from stresses Let a planeelastic wave propagate along the 119909-axis Denoting 119906 and V asthe nonzero components of the displacement vector we set119906 = 119906(119909 119911 119905) V = V(119909 119911 119905) and 119908 = 0 and consequentlywrite for the strain tensor (119890

119894119895)

119890119909119909

=

120597119906

120597119909

119890119911119911=

120597V

120597119911

2119890119909119911= (

120597119906

120597119911

+

120597V

120597119909

) (1)

Let us assume that an external energy flux (eg laserradiation) creates mobile atomic defects in a surface layerThe corresponding defect density profile results in a forcethat may induce strain field in medium Let 119899

119894(119909 119911 119905) be the

density of these defects of the 119895th-type (119894 = 119881 for vacancies(119881-defects) and 119895 = 119868 for interstitials (119868-defects)) We limitour consideration to the case of only one type of atomicdefects (for definiteness 119881-type defects)

The concentration field of atomic defects is dependent ontemperature of themedium One thus needs to know how thelaser irradiation affects the local temperature of the surfaceat the laser spot We will consider here situations wherethe laser irradiation only heats the solid (the light energyabsorbed by the medium is transformed into heat) and thatan equilibrium between laser radiation and the temperaturefield (119879) is reached on time scales much shorter than thecharacteristic time scale of defect density evolution Typicallythe time scale for equilibration between photon absorptionand defect generation is on the order of picoseconds whilethat for defect diffusion is of the order of microsecondsWe also assume that the contribution of thermal strains todeformation fields is negligible compared to lattice dilatationdue to atomic defects and the phase changes and chemicalreactions in the medium are absent

In this paper we will consider the problem of the wavepropagation in an elastic solid irradiated over a large areaby CW or pulsed lasers Furthermore we will assume that

Advances in Condensed Matter Physics 3

the temperature field has reached its equilibrium value Itsevolution is sufficiently slow compared to atomic defectgeneration and can be considered as quasistationary Thesolution of the heat conduction equation for this case is givenby Duley [22]

Taking into account the defect generation the constitu-tive equations of an isotropic nonlocal elastic solid are

120590119894119896 (

x 119905) = int

Ω

120572 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119904119894119896(x1015840 119905) 119889Ω (x1015840)

minus int

Ω

120573 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119899 (x1015840 119905) 120575

119894119896119889Ω (x1015840)

119904119894119896(x1015840 119905) = 120582

0119890 (x1015840 119905) 120575

119894119896+ 21205830119890119894119896(x1015840 119905)

(2)

where 120590119894119896

is the nonlocal stress tensor 119904119894119896

is the classicalstress tensor 120582

0and 120583

0are Lamersquos constants [23] 119890

119894119896=

(119906119894119896+ 119906119896119894)2 is the strain tensor 119890 = 119906

119896119896is the dilatation

and dΩ(x1015840) = 1198891199091015840

11198891199091015840

2 The functions 120572 and 120573 are known

as atom-atom (short-range) and atom-defect (long-range)interaction kernels or moduli of nonlocality which decaysmoothly with distance They reflect the influence of anindependent constitutive variable at a point x1015840 on a dependentconstitutive variable at 119909 Nonlocal moduli are physicalproperties of materials like other physical constants and needto be determined experimentally

The basic difference between classical and nonlocal elas-ticity is in the presence of the volume integrals in (2) whichindicates that the stress at (x 119905) depends on the strain anddefect density at all other points x1015840 of the body at time 119905This signifies that the distant neighbors of a point x have arole to play in the propagation of waves

In absence of body forces the equations ofmotion and theequations of defect density dynamics have the following form

1205881205972119906119894

1205971199052

+ nabla119896120590119894119896= 0 (3)

120597119899

120597119905

= 119866 + 119863nabla2119899 minus

119863

119896119861119879

nabla sdot (119899f) minus 119899120591minus1 (4)

where 120588 is the density of the medium 119866 = 1198660exp(minus119908

119892119896119861119879)

is the thermal-fluctuation generation rate of atomic defects atsites (119866

0is the constant and 119896

119861is the Boltzmann constant)

and 120591 = 1205910exp(119908

119898119896119861119879) is the relaxation time of defects

in the absence 119908119892and 119908

119898are the formation and migration

energies for the defects in crystalsThe first term in the right-hand side of (4) takes into

account laser-induced generation of defects the second termrepresents diffusion with a coefficient 119863 the third termcorresponds to the drift of defects under the influence of theforce f = minusnabla119880int resulting from the nonlocal interactionof defects with an inhomogeneous strain field and thefourth term describes the rate of their disappearance due torecombination processes It is assumed that the generationrate (119866) is spatially uniform

The expression for interaction energy119880int of a defect withthe strain field 119890

119898119898in a nonlocal elastic medium is given by

the formula

119880int = minusint

Ω

120573 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119890 (x1015840) 119889Ω (x1015840) (5)

In the limit when the kernel functions 120572 and 120573 becomeDirac-delta functions (2) reduces to the classical constitutiveequations of local elasticity theory

Thus the final dynamic field equations obtained areintegrodifferential equations for the functions 119906(119909 119905) and119899(119909 119905) It seems to be obvious that a rigorous solution ofsuch equations encounters serious but not insurmountablemathematical difficulties However these equations can bereduced to the partial differential forms under certain con-ditions with physical admissible kernels We consider thatthe long-range internal influences of particles of the bodyare rather rapidly with increasing distance from the particleThus the 2D-kernel functions that characterize the nonlocalinteraction in the 119911-direction may be approximated in termsof delta-like functions as

120572 = 120572 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816

100381610038161003816100381610038161199111015840minus 119911

10038161003816100381610038161003816) = 120572 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816) 120575119899(1199111015840minus 119911)

120573 = 120573 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816

100381610038161003816100381610038161199111015840minus 119911

10038161003816100381610038161003816) = 120573 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816) 120575119899(1199111015840minus 119911)

(6)

Besides the nonlocal kernels 120572 and 120573 are symmetric withrespect to the interchange of the arguments 120585 = x1015840 minus xThen expanding in (3) the functions 119890(x + 120585) and 119899(x + 120585) inTaylor series around 120585 = 0 and neglecting the terms higherthan second-order gradients we can obtain the followingexpression for the nonlocal constitutive relation with strainand defect concentration gradients

120590119894119896 (

x) = (1 + 1198922nabla2) [1205820120575119894119896119890 (x) + 21205830119890119894119896 (x)]

minus (1 + ℎ2nabla2) 1205730120575119894119896119899 (x)

(7)

where ℎ and 119892 are the intrinsic characteristic length scaleparameters characterizing the defect-atom and atom-atominteractions respectively nabla2 = 120597

21205971199092 1205730

= 119870Ω119889is

deformation potential of the defect (Ω119889is the change of the

volume of the medium under formation of a single defectand 119870 is the bulk modulus) The gradient coefficients 119892 andℎ are dependent on the lattice or atomic chain model usedand the interatomic potentials assumed The values of thesecoefficients are taken to be order of interatomic distance Forthe interaction energy 119880int we have from (5) 119880int = minus(1 +

ℎ2nabla2)1205730119890

To obtain the nonzero stress components in terms of thedisplacement components we substitute (1) into (7)

120590119909119909

= (1 + 1198922nabla2) [(1205820+ 21205830)

120597119906

120597119909

+ 1205820

120597V

120597119911

]

minus (1 + ℎ2nabla2) 1205730119899


120590119911119911= (1 + 119892

2nabla2) [(1205820+ 21205830)

120597V

120597119911

+ 1205820

120597119906

120597119909

]

minus (1 + ℎ2nabla2) 1205730119899

120590119909119911= (1 + 119892

2nabla2) 1205830(

120597119906

120597119911

+

120597V

120597119909

)

(8)

Then from (4) two nonzero equations of motion can beobtained

120588

1205972119906

1205971199052=

120597120590119909119909

120597119909

+

120597120590119909119911

120597119911

120588

1205972V

1205971199052=

120597120590119911119909

120597119909

+

120597120590119911119911

120597119911

(9)

Suppose that the stress components and defect densityfield satisfy the boundary conditions

120590119911119911= 120590119911119909= 0

120597119899

120597119911

= 0 for 119911 = 0

119906 V 119899 997888rarr 0 as 119911 997888rarr infin

(10)

We can express the defect density field as 119899 = 1198990+

1198991(1198990

= 119866120591 is a spatially homogeneous solution 1198991is

small nonhomogeneous perturbations) Inserting in (4) thenonlinear terms we get the linearised equation as

1205971198991

120597119905

+

1198991

120591

minus 119863(

12059721198991

1205971199092+

12059721198991

1205971199112)

= minus

11989901205730119863

119896119861119879

[

1205972119890

1205971199092+

1205972119890

1205971199112+ 1198922(

1205974119890

1205971199094+

1205972119890

12059711990921205971199112)]

(11)

The system of (9) and (11) is closely coupled 12059721205971199052 in(9) depends on the defect density field (119899

1) and 120597119899

1120597119905 in

(11) depends on elastic displacement field () The system ofequations thus becomes highly nonlinear

3 Solution of the Problem

To solve the above problem we apply the Fourier integraltransform in the following form

1198991 (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

119899 (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

119906 V (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

V (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

(12)

Substitution of this into (9) gives

120597119911119909

120597119911

minus 119894119902119909119909+ 1205881205962 = 0

120597119911119911

120597119911

minus 119894119902119909119911+ 1205881205962V = 0

1205972119899

1205971199112minus (1199022+

119894120596120591 + 1

1198972

) 119899

=

1198990120573

119896119861119879

(minus1198941199023 + 1199022 1205972V

1205971199112+ 119894119902

1205972

1205971199112minus

1205973V

1205971199113)

(13)

where

119909119909

= minus119894119902 (120582 + 2120583) + 120582

120597V

120597119911

minus 120573119899

119911119911= (120582 + 2120583)

120597V

120597119911

minus 119894119902120582 minus 120573119899

119909119911= 120583(

120597

120597119911

minus 119894119902V)

(14)

where 120596 is the frequency of wave propagation 119902 is the wavenumber and 119897 = radic119863120591 the phase velocity is given by 119888 = 120596

119903119902

and attenuation constant by Γ = minus120596119894 where 120596

119903= Re(120596) and

120596119894= Im(120596) mean respectively the real and imaginary parts

of 120596 120582 = 120582(119902) = 1205820(1 minus 119892

21199022) and 120583 = 120583(119902) = 120583

0(1 minus 119892

21199022)

are the nonlocal elastic moduli 120573 = 120573(119902) = 1205730(1 minus ℎ

21199022) is

the nonlocal constant characterizing lattice deformation dueto atomic defects (119902 119911) V(119902 119911) and 119899(119902 119911) are unknownfunctions (amplitude functions)

The general solutions to (13) are

= 1198861exp (minus120578

1119911) + 119886

2exp (minus120578

2119911) + 119886

3exp (minus120578

3119911)

V = minus

119894119902

1205781

1198861exp (minus120578

1119911) + 119898

21198862exp (minus120578

2119911)

+ 11989831198863exp (minus120578

3119911)

119899 = 11988921198862exp (minus120578

2119911) + 119889

31198863exp (minus120578

3119911)

(15)

where

1205782

1= 1199022minus

1205881205962

120583

(16)

1205782

2and 12057823are the roots of the equation

(1 minus 120575) 1205784minus 1205782[21199022(1 minus 120575) minus 120588

1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]

+(1199022 minus 120588 1205962

120582 + 2120583

)(

119894120596120591 + 1

1198972

+ 1199022) minus 120575119902

4 = 0

(17)

where 119886119894 119894 = 1 2 3 are arbitrary constants 120575 = 119899

01205732(120582 +

2120583)119896119861119879 is the coupling constant of nonlocal defect-strain

interaction and

119898119895=

120578119895

119894119902

119889119895=

120582 + 2120583

120573119894119902

(1205782

1minus 1205782

119895) 119895 = 2 3 (18)


Then 12057822and 12057823are defined from (17) as follows

1205782

2+ 1205782

3= [2119902

2(1 minus 120575) minus 120588

1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]

times (1 minus 120575)minus1

1205782

21205782

3= [(119902

2minus 120588

1205962

120582 + 2120583

)(1199022+

119894120596120591 + 1

1198972

) minus 1205751199024]


(19)

Stress components are obtained by substituting (15) into(8)

119909119909(119902 119911) = minus 2119894119902 (120582 + 120583) 119886

1exp (minus120578

1119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057821198982+ 1205731198892) 1198862exp (minus120578

2119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057831198983+ 1205731198893) 1198863exp (minus120578

3119911)

119911119911(119902 119911) = 2119894119902120583119886

1exp (minus120578

1119911)

+ (119894119902120582 minus (120582 + 2120583) 12057821198982+ 1205731198892) 1198862exp (minus120578

2119911)

+ (minus119894119902120582 minus (120582 + 2120583) 12057831198983+ 1205731198893) 1198863exp (minus120578

3119911)

119909119911(119902 119911) = 120578

1(1 +

1199022

1205782

1

)1198861exp (minus120578

1119911)

+ (1205782+ 119894119902119898

2) 1198862exp (minus120578

2119911)

+ (1205783+ 119894119902119898

3) 1198863exp (minus120578

3119911)

(20)

4 Dispersion Equations of the Wave inan Infinitive Medium

In this section as a special case we consider the generation ofplane harmonic structures in unbounded nonlocal mediumSetting 120578

119895= 0 119895 = 1 2 3 in (16) and (17) we obtain

1205881205962= 120583 (119902) 119902

2 (21)

(1199022+

119894120596120591 + 1

1198972

)[1199022minus 1205962 120588

120582 (119902) + 2120583 (119902)

] minus 120575 (119902) 1199024= 0

(22)

Suppose now that one ignores the influence of the non-local properties of the medium and sets 120582(119902) = 120582

0= const

120583(119902) = 1205830= const and 120575(119902) = 120575

0= const In this case (21)

and (22) reduce to frequency equations

1205962= 1198882

11987901199022 (23a)

(1199022minus 1205962119888minus2

1198710) [1199022+ (119894120596120591 + 1) 119897

minus2] minus 12057501199024= 0 (23b)

(1198881198710

= [(21205830+ 1205820)120588]12 and 119888

1198790= (120583

0120588)12 are the

classical velocities of the longitudinal and shear waves resp)coinciding with the equations derived in the conventional

local theory (see eg [8]) In the nonlocal case the frequencyequation has of course a more complex structure

To explore and delineate the strain and defect generationeffects we will seek solutions of (22) for small values of 120575For 120575 = 0 (22) admits the following solutions 120596(0)

12= plusmn119888119871119902

(acoustical mode) and 120596(0)3

= 119894120591minus1(11989721199022+1) (diffusion mode)

Now for small 120575 ≪ 1 we may write

12059612

= 120596(0)

12+ 120575120596(1)

12+ 1205752120596(2)

12+ sdot sdot sdot

1205963= 120596(0)

3+ 120575120596(1)

3+ 1205752120596(2)

3+ sdot sdot sdot

(24)

Substituting (24) into (22) and equating the coefficients oflike powers of 120575 we finally arrive to

12059612

= 120596(0)

12minus

120575

2

120596(0)

1211990221198972(11990221198972+ 1 minus 119894120596

(0)

12120591)

(11990221198972+ 1)2+ (120596(0)

12120591)

2+ 119874 (120575

2) (25)

1205963= 120596(0)

3minus

11989412057511989721198882

1198711199024120591

1198882

11987111990221205912+ (11990221198972+ 1)2+ 119874 (120575

2) (26)

From (25) we obtain

Re (12059612) = 120596(0)

12[1 minus

120575

2

11990221198972(11990221198972+ 1)

(11990221198972+ 1)2+ (119888119871119902120591)2] (27)

Im (12059612) =

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2+ (119888119871119902120591)2 (28)

If 119888119871119902 ≫ 120591

minus1(11989721199022+ 1) and the viscosity is taken into

account [by adding in (9) the terms 120578119879Δ

and 120578

119871nabla(div

)where 120578

119879= 120578 120578

119871= 41205783 + 120577 120578 and 120577 are the first and second

viscosity coefficients] dispersion (28) describes attenuation ofthe amplitude of acoustic waves

Im (12059612) =

1

2120588

1205781199022+

120575

2120591

(11989721199022+ 1) (29)

For the frequency spectrum of acoustic wave we have

Re (12059612) = 120596(0)

12[1 minus

120575

2120591

(11989721199022+ 1)

119888119871119902

] asymp 120596(0)

12 (30)

Note that the frequency of acoustic waves is hardlychanged But the additional contribution to attenuation coef-ficient of waves arises As attenuation decrement of acousticwaves is proportional to 119902

2 according to (29) short wavesfade fast

If 119888119871119902 ≪ 120591

minus1(11989721199022+ 1)

Re (12059612) = 120596(0)

12(1 minus

120575 11990221198972

2 (11990221198972+ 1)

) (31)

Im (12059612) =

1

2120588

1205781199022+

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2 (32)

In this case there is a softening of frequencies of acousticwaves (instability of frequencies Re(120596

12) rarr 0 Im(120596

12) gt 0)


and this is related to taking into account the generationof atomic defects It is necessary to notice that reductionof frequency occurs not up to zero but up to value 120596

120578=

12057811990222120588 ≪ 119888

119871119902 Equation (32) describes attenuation of the

amplitudes of acoustic wavesAt 120596119894≪ 119888119871119902 from (22) we obtain dispersion relation

(growth rate) Γ = minus Im(120596) of the volume concentration-elastic instability

Γ120591 = 11989721199022(120575 minus 1) minus 1 (33)

or

Γ = 1198631199022[

[

1205750

(1 minus 11989221199022)

2

1 minus ℎ21199022

minus 1]

]

minus

1

120591

(34)

where 1205750= 11989901205732

0(1205820+ 21205830)119896119861119879

Let us introduce the dimensionless growth rate Γ =

Γ1198922119863 the wave number 119902 = 119902119892 the life-time of defects

120591 = 1205911198631198922 and parameters 120574 = 119892

2ℎ2 Dependence (34) in

dimensionless variables takes the form

Γ =

1199022

1 minus 1199022(1199022+ 1205750(1 minus 120574119902

2)

2

minus 1) minus

1

120591

(35)

We consider two situations that differ from each other bythe relation (120574) of the squares of the characteristic lengths 119892and ℎ Figure 1 demonstrates the graphs of the growth ratecalculated using (35) for two values of the ratio parameter120574 = 8 and 10 respectively The following typical valuesof parameters were used in the calculations ℎ = 06 nm1205750

= 10 and 119863 = 4 times 10minus11 cm2 sminus1 Comparison of

the plots presented in Figure 1 shows that the dependencesof the growth rates of the wave number for the two casesqualitatively differ from each other It follows from Figure 1that the growth rate exhibits a single maximum at 119902

119898=

3 times 106 for small values of 120574 and an additional maximum in

the short-wavelength region at 119902 gt 6 times 106 cmminus1 Therefore

depending on the relation between the characteristic lengths119892 and ℎ elastic-concentration structures have one or twomaximum growth rates

Figure 2 shows the dependences Γ(119902) of the incrementon the wave number for two values of the control parameter1205750= 5 and 15 when the ratio parameter is 120574 = 8 It follows

from this figure that the growth rate has a single maximumnot far above the threshold However at sufficiently highconcentrations of the defects exceeding the threshold appearsan additional maximum in the short-wavelength rangeTherefore as in the previous case two gratings have themaximum growth rates

The analytical expression for the maximum of the growthrates can be obtained provided that the nonlocal parametersare of the same order in (34) (ie 119892 asymp ℎ) Then using (34)we have

Γ (119902) = 1198631199022[

119866

119866119888

(1 minus 11989221199022) minus 1] minus

1

120591

(36)

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)

Nondimensional wave number (q)

Figure 1 Dependences of the growth rates on the wave numberfor two values of the ratio parameter 120574 8 (solid line) 10 (dot line)Control parameter 120575

0= 10

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)


Figure 2 Dependences of the growth rate on the wave number fortwo values of the control parameter 120575

0 5 (solid line) 15 (dot line)

Ratio parameter 120574 = 80

where

119866119888= 119896119861119879120588 1198882

119871(1205911205732

0)

minus1

(37)

is the critical value of the control parameter (defect genera-tion rate) So putting values into (37) for molybdenum 120573

0=

20 eV 1205881198882119871= 1012 erg cmminus3 119879 = 10

3 K and 120591 = 10minus3 s one

can get the estimate 119866119888= 1024 sminus1 cmminus3


The dispersion curve Γ(119902) has a maximum Γ119898

at 119902 =

119902119898 which determines the spatial scale 119889latt = 2120587119902

119898of the

dominant CS structures

119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)

The maximum growth rate is

Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888

minus 120591minus1 (39)

It is seen from (39) that the development of the elastic-concentration instability with the self-organization of CSstructures in the volume of the solids occurs in a thresholdmanner when the defect generation rate exceeds the criticalvalue 119866 gt 119866

119888 At the elastic-concentration instability

threshold (119866 rarr 119866119888) the period tends to infinity 119889latt rarr

infin At high values of generation rates 119866 ≫ 119866119888 the period

asymptotically tends to its minimal value 119889latt = radic8120587119892Near the threshold 119866 rarr 119866

119888the time of formation of the

CS structures Γminus1119898

rarr infin that is critical slowing down takesplace which is characteristic of the phase transitions of thesecond kind

The critical value of the generation rate derived hereis directly proportional to recombination rate (120591minus1) andinversely proportional to deformational potentials (120573

0) of

defects Putting 120591 = 1205910exp(119908

119898119896119861119879) into (37) for the

generation rate we get

119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)

One notes that this dependence is very rapid which islargely due to variation of the defect relaxation time withtemperature

From (38) it follows that the period of the volumeordered CS structures is proportional to the characteristiclength ℎ of the defect interaction with the crystal-latticeatoms which lies in the obtained nanometer range Astemperature is increased the period grows for a particularvalue of ℎ For a particular value of 119879 the period decreaseswith an increasing generation rate (119866)

A periodic structure of vacancy clusters in the bulk of asolid irradiated with nitrogen ions was first observed in 1971by Evans in pure molybdenum [24] The medium tempera-ture was 119879 = 1040∘K Subsequently the same effect had beenobserved in other metals such as aluminum tungsten nickeland niobium [25 26] Vacancy cluster sizes in the lattice were2ndash4 nm and values of the lattice parameter change in therange of (20ndash60) nmAt119892 = 15 nm (for119866 = 11119866

119888) from (38)

we obtain an estimate for the period of the volume structure119889latt = 44 nmThis is in qualitative agreement with the above-mentioned experimentally observed interval

Note that the linear theory considered in this sectiondescribes early stage of the development of an elastic-concentration instability only However the nature of gener-ated ordered volume structures (due to an instability) and theamplitudes of these structures as functions on material andirradiation conditions can only be determined by consideringthe influence of elastic nonlinear effects in the model

5 Dispersion Equations of the SurfaceConcentration-Strain Structures

Substituting expressions (20) into boundary conditions (10)and taking into account expressions (18) we find a set of threelinear algebraic equations

211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)

The condition of existence of a nontrivial solution of thissystem yields

(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578

3are given by (16) and (19)

Equation (42) shows dispersive character of the Rayleighwaves propagating in an elastic half-space This dispersivecharacter of these waves propagation arises due to atomicdefect generation in the medium with long-range interac-tions

Note that the form of this dispersion equation is identicalto the corresponding equation obtained in the context ofconcentration-elastic coupled theory using classical consti-tutive equations of motion though each of 120578

1 1205782 and 120578

3

represents different expressions for classical elasticity andnonlocal elasticity

Equation (21) is generalized form of the classical localRaleigh equations The assumption of vanishing defect gen-eration in this equation gives us exactly the same results asthose obtained by Nowinski [17] for solids with long-rangeinteractions that is

4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)

6 Solution of Dispersion Equation

A general discussion of (42) presents considerable difficul-ties The structure of (42) is identical to the structure of thedispersion equation for DEI considered in [8] This equationcan be readily solved numerically or graphically but in somecases of interest it can also be solved analyticallyWe shall nowconsider the results for some limiting cases We can rewrite(42) as

(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783


Equation (45) has solutions describing qualitatively dif-ferent types of instability (1) instability of frequencies ofsurface acoustic waves and (2) generation of ordered surface(static) concentration-strain structures

If the coupling term is small (120575 ≪ 1) the expressions thatappear in the dispersion equation (25) to a first approxima-tion then read

119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)

Thenwemay express the dispersion equation in the followingform

119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)

For 120575 = 0 we obtain the nonlocal elastic solution given byNowinski [17] Introducing the dimensionless variable 120585 =

(1205961198881119902)2 we may write the dispersion equation (47) in the

following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)

Now we may consider that 120576 is the increment of the value of120585 due to 120575 = 0 Then (49) can be rewritten in the form of

119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)

where 120585119877is a root of the equation 119877(120585

119877) = 0

Assuming that 120576 ≪ 1 we may expand both sides of(50) into a Taylor series in the vicinity of the point 120585 = 120585

119877

Retaining only the first two terms we obtain

120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)

The real part of 120576 characterizes the change of the phasevelocity and its imaginary part (120576

119894) defines the attenuation

constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)

This equation gives the following expressions

120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877

Then for the phase velocity we have

119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)

Since 120575 gt 0 in this case there is a softening of frequencies ofacoustic waves (instability of frequencies120596

119903rarr 0 and120596

119894gt 0)

and this is related to taking into account the generation ofatomic defects Equation (53b) describes attenuation of theamplitudes of acoustic waves

Expanding 1205781and 120578

2in powers of the small parameter

120596119894119888119879sdot119871

≪ 1 (120596119903= 0) we obtain from (21) the expression

for the growth rate Γ = minus120596119894of the surface periodical static

structures in the form

Γ = 1198631199022[

[

120575

(1 minus 120577minus1)

(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879

Introducing the dimensionless growth rate Γ = Γ1198922119863

the wave number 119902 = 119902119892 the life-time of defects 120591 = 1205911198631198922

and parameters 120601 = 120575(1 minus 120577minus1) 120574 = 119892

2ℎ2 dependence (34)

can be represented as

Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2

minus 1) minus 120591minus1 (56)

The analysis shows that the growth rate exhibits a singlemaximum at 119902 = 119902

119898for small values of ratio parameter

(120574) and an additional maximum (119902119888) in the short-wavelength

region for large values of 120574 Thus depending on the rela-tion between the characteristic lengths 119892 and ℎ elastic-concentration structures can have one or two (with 119902 = 119902

119898

and 119902 = 119902119888) maximum growth rates

Besides the growth rate has a single maximum not farabove the threshold However at sufficiently high concentra-tions of the defects exceeding the threshold appears an addi-tional maximum in the short-wavelength rangeTherefore asin the previous case two gratings have the maximum growthrates

The obvious expression for the maximum of the growthrates can be obtained if the scale coefficients (119892 ℎ) are of thesame order in (34)

Γ = 1199022[120601 (1 minus 119902

2) minus 1] minus 120591

minus1 (57)

The growth rate Γ(119902) reaches maximum Γ119898at

119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)

is the critical value of the control parameter at exceedingof which the self-organization of surface-ordered CS struc-tures becomes possible due to development of the elastic-concentration instability As a result the surface deformation


lattice and the corresponding relief modulation with theperiod 119889(119904)latt grow in time on the surface as well as the defectconcentration lattice (defect accumulations at the extrema ofthe surface relief)

The spatial scale of surface SC structures with wavenumber 119902

119898is given by

119889(119904)

latt =2120587

119902119898

= radic8120587119892radic

119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2


Near the threshold 119866 rarr 119866119888the time of formation of the CS

structures Γminus1119898

rarr infin that is critical slowing down takes placewhich is characteristic of the phase transitions of the secondkind

The period of the ordered CS structures formed on thesurface is proportional to the characteristic length ℎ of thenonlocal defect-atom interaction which lies in the obtainednanometre range As temperature is increased the periodgrows for a particular value of ℎ For a particular value of 119879the period decreases with an increasing generation rate (119866)

Thus at certain conditions concentration-elastic instabili-ties in a system of atomic defects in unbounded condensedmedia as well as on the free surface of a solid half-spacecan be developed We have observed that if the pumpparameter is above the critical value due to concentration-elastic instability coupled strain-defect periodic nanometersized structures on the surface of solids arise A mechanismon the development of the instability is due to the couplingbetween defect dynamics and the elastic field of the solidsLaser radiation (or in general a flux of particles) generateshigh concentrations of mobile atomic defects in the surfacelayer of the irradiatedmaterial When a fluctuation harmonicof the elastic deformation field appears in a medium becauseof the generation of atomic defects the strain-induced driftof atomic defects occurs This is a consequence of nonlocaldefect-strain interaction The strain-induced flux of defectsgives rise to periodic spatial-temporal fields of the defectconcentration The redistribution of defects creates forcesproportional to their gradients These forces lead in turnto additional growth of strain fluctuations When a certaincritical value of the defect density or rate of defect generationis exceeded diffusion-elastic instabilities develop as a resultof positive feedback which result in the formation of orderedCS structures

As an example let us consider the formation of periodicsurface CS structures in laser-irradiated semiconductors (inparticular CdTe) To evaluate the concentration of generatedlattice defects (119899

0) we consider here conditions when the

duration of a laser pulse (120591Las) exceeds the defect-relaxationtime (120591) In this case the density of defects on the surface ofthe solid reaches a steady-state value

1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)

is the defect generation rate 1198790is the steady-state value of

the temperature field on the surface and 119866(0) and 120591

0are

constants If 119868Las = const (uniform irradiation) and theoptical absorption length (120572minus1Las) is sufficiently less than heatdiffusion length 119897

119879= (120594120591Las)

12 the maximum temperaturerise at the substrate surface owing to the laser pulse actionmay be evaluated [22] as

1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)

where 119877 is the reflectivity coefficient 120582119879is the thermal

conductivity coefficient and 120594 is the thermal diffusioncoefficient So putting values into (64) for CdTe (120582

119879=

005Wcm∘K120594 = 3times10minus3 cm2s 119868

0= 12times10

7Wcm2 pulseduration 2times10minus8 s and119877 = 04) one can get119879

0= 12times10

3 ∘KThen taking Ω

119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908

1198890= 1 eV a value of 2times1019 1cm3may be estimated for

the critical density of defect concentration (119899119888= 119866(119904)

119888120591) which

is several orders of magnitude less than the concentrationof the host atoms and shows that this mechanism of theformation of ordered structures may be realized on practiceFor the period (119889latt) of the resultant surface CS structure wehave an estimate of 71 nm which follows from expression(60) (for typical values of parameters 120588 = 585 gcm3 120573

0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm

119863 = 10minus6 cm2s and 120591 = 3 times 10

minus9 s) The maximum growthincrement of the instability is Γ

119898= 6 times 10

8 sminus1 the instabilityincrement exceeds the reciprocal of the duration of a laserpulse acting on the surface of a solid (Γ

119898120591Las = 12 ≫ 1)

The appearance of periodic nanostructures of the surfacerelief was detected experimentally [27] when the surface ofthe p-CdTe semiconductor crystal was irradiated by singlepulses of the multimode ruby laser (pulse duration 120591Las =

2 times 10minus8 s) with a homogeneous distribution of intensity The

period of the relief structure (with lateral size of the nanoclus-ters prop (19ndash23) nm) was 74 nm which is in agreement withthe predictions of our theory

7 Conclusions

Based on the nonlocal elastic theory which accounts for thesmall-scale effect by assuming the stress at a reference pointas a function of the strain at every point in the body thegoverning equations for the elastic displacement vector anddefect density fields in an isotropic laser-excited solid havebeen derived The defect-density dynamics are controlled bythe generation and annihilation processes and also the strain-stimulated transport of defects We have provided exactformal solutions for the displacements and defect densityfields in an elastic solid half-space A linear analysis is usedto obtain dispersion equations of surface waves through thesolution of three coupled integropartial differential equa-tions by assuming that the surface of the solids is stress-free A procedure for determining the phase velocity and


the attenuation (amplification) factor has been discussedTheproposed analysis is applied for the special case of very shortwavelengths The phase velocity and attenuation constants ofthe waves get modified due to the defect subsystem of thesolid We have shown that the dispersion equations predicttwo types of the instability (1) instability of frequencies ofacoustic waves and (2) generation of ordered (surface orvolume) structures

Weobserved that the small-scale effect plays an importantrole in the dispersion behaviors for larger wave numbersWe have demonstrated that due to the nonlocal character ofthe defect-atom and atom-atom interactions the dispersioncurves of the instability have two maxima sufficiently farabove the instability threshold When the regular surfacerelief is generated two maxima of the growth rate must giverise to two scales of the surface relief These two scales ofthe relief are the characteristic of the surface nanostructureself-organization upon both laser and ion-beam irradiationof semiconductors

In the case of the laser molecular-beam epitaxy (MBE)and the laser-controlled deposition process of atoms onsubstrates the extrema (minima or maxima) of the surfacerelief formed by the CS structure serve as the nucleationand growth centers of the nanoparticles Similarly in theprocesses of multipulse laser etching the rapid materialremoval also takes place on the extremes of the CS fieldwhere the removal rate takes maximum values As a resulton the surface of substrate size distribution functions ofthe nanoscale nonhomogeneities having two maxima can beformed A transformation of the unimodal distribution to thebimodal (with two maxima) distribution upon a variationin the irradiation regime (control parameter) is typical ofboth theMBE and the laser-induced surface nanostructuringof metals and semiconductors Formation of bimodal metal(Au) nanoparticle size distribution function upon the pulsedlaser deposition was detected experimentally in [28]

The results obtained in this work are expected to be help-ful in designing the nanostructures in small-scale physicaldevices

References

[1] F Mirzade ldquoNonlinear strain waves interacting with laser-induced carries of the local disorderrdquo in Laser Technologiesof Materials Treatment V Y Panchenko Ed pp 220ndash277Fizmatlit Moscow Russia 2009 Russian

[2] F Mirzade ldquoElastic wave propagation in a solid layer with laser-induced point defectsrdquo Journal of Applied Physics vol 110 no 6Article ID 064906 2011

[3] F Mirzade ldquoA model for the propagation of strain solitarywaves in solids with relaxing atomic defectsrdquo Journal of AppliedPhysics vol 103 no 4 Article ID 044904 2008

[4] F Mirzade ldquoInfluence of atomic defect generation on the prop-agation of elastic waves in laser-excited solid layersrdquo Physica Bvol 406 no 24 pp 4644ndash4651 2011

[5] A M Kosevich Physical Mechanics of Real Crystals NaukaMoscow Russia 1986

[6] F Mirzade V Y Panchenko and L A Shelepin ldquoLaser controlof processes in solidsrdquoUspechi Phyzicheskih Nauk vol 39 no 1pp 3ndash30 1996

[7] F Mirzade ldquoConcentration-elastic instabilities in a solid half-spacerdquo Physica Status Solidi B vol 246 no 7 pp 1597ndash16032009

[8] F Mirzade ldquoOn diffusion-elastic instabilities in a solid half-spacerdquo Physica B vol 406 no 1 pp 119ndash124 2011

[9] D Walgraef and N Ghoniem ldquoModeling laser-induced defor-mation patterns nonlinear effects and numerical analysisrdquoJournal of Computer-Aided Materials Design vol 6 no 2 pp323ndash335 1999

[10] F Mirzade ldquoFinite-amplitude strain waves in laser-excitedplatesrdquo Journal of Physics Condensed Matter vol 20 no 27Article ID 275202 2008

[11] A C Eringen Nonlocal Continuum Field Theories SpringerNew York NY USA 2002

[12] A I Kunin Elastic Media with Microstructure Springer NewYork NY USA 1982

[13] A C Eringen ldquoOn differential equations of nonlocal elasticityand solutions of screw dislocation and surface wavesrdquo Journalof Applied Physics vol 54 no 9 pp 4703ndash4710 1983

[14] J N Reddy ldquoNon-local nonlinear formulations for bending ofclassical and shear deformation theories of beams and platesrdquoInternational Journal of Engineering Science vol 48 no 11 pp1507ndash1518 2010

[15] A Alibeigloo ldquoFree vibration analysis of nano-plate usingthree-dimensional theory of elasticityrdquo Acta Mechanica vol222 no 1-2 pp 149ndash159 2011

[16] B Arash and Q Wang ldquoA review on the application ofnonlocal elastic models in modeling of carbon nanotubes andgraphenesrdquo Computational Materials Science vol 51 no 1 pp303ndash313 2012

[17] J L Nowinski ldquoOn non-local theory of wave propagation inelastic platesrdquo ASME Journal of Applied Mechanics vol 51 no3 pp 608ndash613 1984

[18] J L Nowinski ldquoOn the propagation of thermoelastic wavesin media with long-range cohesive forcesrdquo Journal of ThermalStresses vol 10 no 1 pp 17ndash27 1987

[19] K L Pan and J Fang ldquoInteraction energy of dislocation andpoint defect in bcc ironrdquo Radiation Effects and Defects in Solidsvol 139 no 2 pp 147ndash154 1996

[20] K L Pan ldquoAn image force theorem for a screw dislocation neara crack in non-local elasticityrdquo Journal of Physics D vol 27 no2 pp 344ndash346 1994

[21] K-L Pan ldquoInteraction of a dislocation with a surface crack innonlocal elasticityrdquo International Journal of Fracture vol 69 no4 pp 307ndash318 1995

[22] WW Duley Laser Processing and Analysis of Materials PlenumPress New York NY USA 1983

[23] L D Landau and E M LifshitzTheory of Elasticity PergamonPress Oxford UK 3rd edition 1986

[24] J H Evans A W MacLean A A Ismail and D LoveldquoConcentrations of plasma testosterone in normal men duringsleeprdquo Nature vol 229 pp 261ndash262 1971

[25] V K Sikka and J Moteff ldquoDamage in neutron-irradiated moly-bdenumrdquo Journal of Nuclear Materials vol 54 no 2 pp 325ndash345 1974


[26] V F Zelensky IMNekludov and L S Ozhigov Some Problemsin the Physics of Radiation Damage in Materials NaukovaDumka Kiev Ukraine 1979

[27] A Baidullaeva A B Vlasenko L F Cuzan et al ldquoFormation ofnanosize structures on a surface of p-CdTe crystals due to theruby laser single pulse irradiationrdquo Semoconductors vol 39 pp1064ndash1067 2005

[28] J Gonzalo A Perea D Babonneau et al ldquoCompeting processesduring the production of metal nanoparticles by pulsed laserdepositionrdquo Physical Review B vol 71 no 12 Article ID 1254208 pages 2005

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Limitations of the classical elasticity theory are alsodemonstrated in the study of the formation of coupled strain-defect nanometer sized ordered structures (short-range CSstructures) on the surface of laser-irradiated solids andmechanical behavior of microstructured materials becausetheir behavior is characterized by nonlocal stresses and theexistence of an internal length scale [11 12] The accurateanalysis of dynamic behaviors of these structures cannotbe correctly described by classical local elasticity theoryConducting experiments with nanoscale size materials isfound to be difficult and expensive Therefore develop-ment of appropriate nonlocal elasticity mathematical modelsfor nanostructure formation and nanomaterials is of greatimportance

In contrast to local approach of zero-range internalinteractions the nonlocal elasticity theory originated anddeveloped in the last four decades postulates that the stressat a point 119909 in a body depends not only on the strainat point 119909 but also on those at all other points of thebody Various nonlocal theories of linear elasticity have beenproposed to describe the scale effects on the characteristicsof the vibration and elastic wave propagation in the above-mentioned submicro- or nanosized structures The basicconstitutive equations and governing equations of linearnonlocal elasticity were derived by Eringen [11] and Kunin[12] Later the nonlocal theories have been applied for theanalysis of micro- and nanoscale plate-like structures inwhich the small-scale effects become significant In [13] Erin-gen considered vibration behavior of a nanoplate by usinglinear theory of nonlocal continuummechanics By using [11]Reddy [14] reformulated the classical and shear deformationbeam and plate theoriesThe nonlocal scale influences on thewave dispersion properties of the nanoplates are discussed indetail in [15] Vibration characteristics of nanoplates basedon three-dimensional theory of elasticity employing nonlocalcontinuummechanics have been discussed in [15] A reviewof some other applications of nonlocal elasticity theoriesfor nanostructures can be found in [16] Propagation oflongitudinal elastic and thermoelastic waves in an isotropichomogeneous infinite medium with long-rang interactionshas been studied by Nowinski [17 18] Other advances havebeen made by the application of nonlocal elasticity to suchfields as the solid defects [19 20] and fracture mechanics [21]

The present paper is concerned with the nonlocal elastic-ity theory of laser-excited solid half-space withmobile atomicpoint defects We summarise the theory formulated in [8] toanalyze the effects of nonlocal atom-atom and atom-defectinteractions on the surface wave propagation in solids withdefect generation The changes in defect concentration arecontrolled by the (i) generation of defects by laser irradia-tion (ii) their diffusion and (iii) strain-induced diffusionDispersion equation that governs the propagation of elastic-concentration waves has been derived by solving a system ofcoupled partial differential equations Some limiting cases ofthe dispersion equations are considered For a small value ofcoupled parameter obvious expressions for the phase velocityand the attenuation (amplification) factor of surface elastic-concentration waves are determined It is found that bothphase velocity and attenuation coefficient are to be influenced

by the presence of defect generation We also obtain thatat certain conditions concentration-elastic instabilities withthe formation of regular nanosized structures in a system ofatomic defects on the surface of the solids can be developedWe demonstrate that due to the nonlocal character of thedefect-atom and atom-atom interactions the dispersive curveof the instability has twomaxima As a result size distributionfunctions of the surface nanoscale nonhomogeneities havingtwomaxima can be formedThe results of some earlier worksare also deduced from the present formulation To our knowl-edge the problem in its present form was not investigatedbefore

2 Governing Equations

In this section we give basic 2D equations governing thedefect density and elastic displacement fields based on thenonlocal constitutive relations of the elasticity theory Con-sider an isotopic elastic semi-infinite medium occupying aregion minusinfin lt 119909 lt infin 0 lt 119911 lt infin in a rectangular Cartesiancoordinate system 119874 119909 119910 119911 where the origin 119874 is situated atany point on the plane boundary and 119874119911 points verticallydownwards that is towards the bulk of the medium Thesurface 119911 = 0 is supposed to be free from stresses Let a planeelastic wave propagate along the 119909-axis Denoting 119906 and V asthe nonzero components of the displacement vector we set119906 = 119906(119909 119911 119905) V = V(119909 119911 119905) and 119908 = 0 and consequentlywrite for the strain tensor (119890

119894119895)

119890119909119909

=

120597119906

120597119909

119890119911119911=

120597V

120597119911

2119890119909119911= (

120597119906

120597119911

+

120597V

120597119909

) (1)

Let us assume that an external energy flux (eg laserradiation) creates mobile atomic defects in a surface layerThe corresponding defect density profile results in a forcethat may induce strain field in medium Let 119899

119894(119909 119911 119905) be the

density of these defects of the 119895th-type (119894 = 119881 for vacancies(119881-defects) and 119895 = 119868 for interstitials (119868-defects)) We limitour consideration to the case of only one type of atomicdefects (for definiteness 119881-type defects)

The concentration field of atomic defects is dependent ontemperature of themedium One thus needs to know how thelaser irradiation affects the local temperature of the surfaceat the laser spot We will consider here situations wherethe laser irradiation only heats the solid (the light energyabsorbed by the medium is transformed into heat) and thatan equilibrium between laser radiation and the temperaturefield (119879) is reached on time scales much shorter than thecharacteristic time scale of defect density evolution Typicallythe time scale for equilibration between photon absorptionand defect generation is on the order of picoseconds whilethat for defect diffusion is of the order of microsecondsWe also assume that the contribution of thermal strains todeformation fields is negligible compared to lattice dilatationdue to atomic defects and the phase changes and chemicalreactions in the medium are absent

In this paper we will consider the problem of the wavepropagation in an elastic solid irradiated over a large areaby CW or pulsed lasers Furthermore we will assume that




120590119894119896 (

x 119905) = int

Ω

120572 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119904119894119896(x1015840 119905) 119889Ω (x1015840)

minus int

Ω

120573 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119899 (x1015840 119905) 120575

119894119896119889Ω (x1015840)

119904119894119896(x1015840 119905) = 120582

0119890 (x1015840 119905) 120575

119894119896+ 21205830119890119894119896(x1015840 119905)

(2)

where 120590119894119896



0and 120583


119894119896=



and dΩ(x1015840) = 1198891199091015840

11198891199091015840





1205881205972119906119894

1205971199052

+ nabla119896120590119894119896= 0 (3)

120597119899

120597119905

= 119866 + 119863nabla2119899 minus

119863

119896119861119879



119892119896119861119879)




and 120591 = 1205910exp(119908








the formula


Ω

120573 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119890 (x1015840) 119889Ω (x1015840) (5)



120572 = 120572 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816

100381610038161003816100381610038161199111015840minus 119911

10038161003816100381610038161003816) = 120572 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816) 120575119899(1199111015840minus 119911)

120573 = 120573 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816

100381610038161003816100381610038161199111015840minus 119911

10038161003816100381610038161003816) = 120573 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816) 120575119899(1199111015840minus 119911)

(6)


120590119894119896 (

x) = (1 + 1198922nabla2) [1205820120575119894119896119890 (x) + 21205830119890119894119896 (x)]

minus (1 + ℎ2nabla2) 1205730120575119894119896119899 (x)

(7)


21205971199092 1205730

= 119870Ω119889is



ℎ2nabla2)1205730119890


120590119909119909

= (1 + 1198922nabla2) [(1205820+ 21205830)

120597119906

120597119909

+ 1205820

120597V

120597119911

]

minus (1 + ℎ2nabla2) 1205730119899


120590119911119911= (1 + 119892

2nabla2) [(1205820+ 21205830)

120597V

120597119911

+ 1205820

120597119906

120597119909

]

minus (1 + ℎ2nabla2) 1205730119899

120590119909119911= (1 + 119892

2nabla2) 1205830(

120597119906

120597119911

+

120597V

120597119909

)

(8)


120588

1205972119906

1205971199052=

120597120590119909119909

120597119909

+

120597120590119909119911

120597119911

120588

1205972V

1205971199052=

120597120590119911119909

120597119909

+

120597120590119911119911

120597119911

(9)


120590119911119911= 120590119911119909= 0

120597119899

120597119911

= 0 for 119911 = 0


(10)


1198991(1198990



1205971198991

120597119905

+

1198991

120591

minus 119863(

12059721198991

1205971199092+

12059721198991

1205971199112)

= minus

11989901205730119863

119896119861119879

[

1205972119890

1205971199092+

1205972119890

1205971199112+ 1198922(

1205974119890

1205971199094+

1205972119890

12059711990921205971199112)]

(11)


1) and 120597119899

1120597119905 in




1198991 (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

119899 (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

119906 V (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

V (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

(12)


120597119911119909

120597119911

minus 119894119902119909119909+ 1205881205962 = 0

120597119911119911

120597119911

minus 119894119902119909119911+ 1205881205962V = 0

1205972119899

1205971199112minus (1199022+

119894120596120591 + 1

1198972

) 119899

=

1198990120573

119896119861119879

(minus1198941199023 + 1199022 1205972V

1205971199112+ 119894119902

1205972

1205971199112minus

1205973V

1205971199113)

(13)

where

119909119909

= minus119894119902 (120582 + 2120583) + 120582

120597V

120597119911

minus 120573119899

119911119911= (120582 + 2120583)

120597V

120597119911

minus 119894119902120582 minus 120573119899

119909119911= 120583(

120597

120597119911

minus 119894119902V)

(14)


119903119902


119903= Re(120596) and


of 120596 120582 = 120582(119902) = 1205820(1 minus 119892

21199022) and 120583 = 120583(119902) = 120583

0(1 minus 119892

21199022)


21199022) is



= 1198861exp (minus120578

1119911) + 119886

2exp (minus120578

2119911) + 119886

3exp (minus120578

3119911)

V = minus

119894119902

1205781

1198861exp (minus120578

1119911) + 119898

21198862exp (minus120578

2119911)

+ 11989831198863exp (minus120578

3119911)

119899 = 11988921198862exp (minus120578

2119911) + 119889

31198863exp (minus120578

3119911)

(15)

where

1205782

1= 1199022minus

1205881205962

120583

(16)

1205782



1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]

+(1199022 minus 120588 1205962

120582 + 2120583

)(

119894120596120591 + 1

1198972

+ 1199022) minus 120575119902

4 = 0

(17)


01205732(120582 +


interaction and

119898119895=

120578119895

119894119902

119889119895=

120582 + 2120583

120573119894119902

(1205782

1minus 1205782

119895) 119895 = 2 3 (18)



1205782

2+ 1205782

3= [2119902

2(1 minus 120575) minus 120588

1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]


1205782

21205782

3= [(119902

2minus 120588

1205962

120582 + 2120583

)(1199022+

119894120596120591 + 1

1198972

) minus 1205751199024]


(19)


119909119909(119902 119911) = minus 2119894119902 (120582 + 120583) 119886

1exp (minus120578

1119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057821198982+ 1205731198892) 1198862exp (minus120578

2119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057831198983+ 1205731198893) 1198863exp (minus120578

3119911)

119911119911(119902 119911) = 2119894119902120583119886

1exp (minus120578

1119911)

+ (119894119902120582 minus (120582 + 2120583) 12057821198982+ 1205731198892) 1198862exp (minus120578

2119911)


3119911)

119909119911(119902 119911) = 120578

1(1 +

1199022

1205782

1

)1198861exp (minus120578

1119911)

+ (1205782+ 119894119902119898

2) 1198862exp (minus120578

2119911)

+ (1205783+ 119894119902119898

3) 1198863exp (minus120578

3119911)

(20)



119895= 0 119895 = 1 2 3 in (16) and (17) we obtain

1205881205962= 120583 (119902) 119902

2 (21)

(1199022+

119894120596120591 + 1

1198972

)[1199022minus 1205962 120588

120582 (119902) + 2120583 (119902)

] minus 120575 (119902) 1199024= 0

(22)


0= const

120583(119902) = 1205830= const and 120575(119902) = 120575



1205962= 1198882

11987901199022 (23a)

(1199022minus 1205962119888minus2

1198710) [1199022+ (119894120596120591 + 1) 119897

minus2] minus 12057501199024= 0 (23b)

(1198881198710

= [(21205830+ 1205820)120588]12 and 119888

1198790= (120583

0120588)12 are the




12= plusmn119888119871119902




12059612

= 120596(0)

12+ 120575120596(1)

12+ 1205752120596(2)

12+ sdot sdot sdot

1205963= 120596(0)

3+ 120575120596(1)

3+ 1205752120596(2)

3+ sdot sdot sdot

(24)


12059612

= 120596(0)

12minus

120575

2

120596(0)

1211990221198972(11990221198972+ 1 minus 119894120596

(0)

12120591)

(11990221198972+ 1)2+ (120596(0)

12120591)

2+ 119874 (120575

2) (25)

1205963= 120596(0)

3minus

11989412057511989721198882

1198711199024120591

1198882

11987111990221205912+ (11990221198972+ 1)2+ 119874 (120575

2) (26)

From (25) we obtain

Re (12059612) = 120596(0)

12[1 minus

120575

2

11990221198972(11990221198972+ 1)

(11990221198972+ 1)2+ (119888119871119902120591)2] (27)

Im (12059612) =

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2+ (119888119871119902120591)2 (28)

If 119888119871119902 ≫ 120591



and 120578

119871nabla(div

)where 120578

119879= 120578 120578



Im (12059612) =

1

2120588

1205781199022+

120575

2120591

(11989721199022+ 1) (29)


Re (12059612) = 120596(0)

12[1 minus

120575

2120591

(11989721199022+ 1)

119888119871119902

] asymp 120596(0)

12 (30)



If 119888119871119902 ≪ 120591

minus1(11989721199022+ 1)

Re (12059612) = 120596(0)

12(1 minus

120575 11990221198972

2 (11990221198972+ 1)

) (31)

Im (12059612) =

1

2120588

1205781199022+

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2 (32)


12) rarr 0 Im(120596

12) gt 0)



120578=

12057811990222120588 ≪ 119888




Γ120591 = 11989721199022(120575 minus 1) minus 1 (33)

or

Γ = 1198631199022[

[

1205750

(1 minus 11989221199022)

2

1 minus ℎ21199022

minus 1]

]

minus

1

120591

(34)

where 1205750= 11989901205732

0(1205820+ 21205830)119896119861119879



120591 = 1205911198631198922 and parameters 120574 = 119892



Γ =

1199022

1 minus 1199022(1199022+ 1205750(1 minus 120574119902

2)

2

minus 1) minus

1

120591

(35)




119898=







Γ (119902) = 1198631199022[

119866

119866119888


1

120591

(36)

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)



0= 10

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)





where

119866119888= 119896119861119879120588 1198882

119871(1205911205732

0)

minus1

(37)


0=

20 eV 1205881198882119871= 1012 erg cmminus3 119879 = 10





at 119902 =


119898of the


119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)


Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888











0) of


119898119896119861119879) into (37) for the


119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)




119888) from (38)





211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)


(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578




1 1205782 and 120578

3



4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)



(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783




119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120590119894119896 (

x 119905) = int

Ω

120572 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119904119894119896(x1015840 119905) 119889Ω (x1015840)

minus int

Ω

120573 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119899 (x1015840 119905) 120575

119894119896119889Ω (x1015840)

119904119894119896(x1015840 119905) = 120582

0119890 (x1015840 119905) 120575

119894119896+ 21205830119890119894119896(x1015840 119905)

(2)

where 120590119894119896



0and 120583


119894119896=



and dΩ(x1015840) = 1198891199091015840

11198891199091015840





1205881205972119906119894

1205971199052

+ nabla119896120590119894119896= 0 (3)

120597119899

120597119905

= 119866 + 119863nabla2119899 minus

119863

119896119861119879



119892119896119861119879)




and 120591 = 1205910exp(119908








the formula


Ω

120573 (

10038161003816100381610038161003816x1015840 minus x1003816100381610038161003816

1003816) 119890 (x1015840) 119889Ω (x1015840) (5)



120572 = 120572 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816

100381610038161003816100381610038161199111015840minus 119911

10038161003816100381610038161003816) = 120572 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816) 120575119899(1199111015840minus 119911)

120573 = 120573 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816

100381610038161003816100381610038161199111015840minus 119911

10038161003816100381610038161003816) = 120573 (

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816) 120575119899(1199111015840minus 119911)

(6)


120590119894119896 (

x) = (1 + 1198922nabla2) [1205820120575119894119896119890 (x) + 21205830119890119894119896 (x)]

minus (1 + ℎ2nabla2) 1205730120575119894119896119899 (x)

(7)


21205971199092 1205730

= 119870Ω119889is



ℎ2nabla2)1205730119890


120590119909119909

= (1 + 1198922nabla2) [(1205820+ 21205830)

120597119906

120597119909

+ 1205820

120597V

120597119911

]

minus (1 + ℎ2nabla2) 1205730119899


120590119911119911= (1 + 119892

2nabla2) [(1205820+ 21205830)

120597V

120597119911

+ 1205820

120597119906

120597119909

]

minus (1 + ℎ2nabla2) 1205730119899

120590119909119911= (1 + 119892

2nabla2) 1205830(

120597119906

120597119911

+

120597V

120597119909

)

(8)


120588

1205972119906

1205971199052=

120597120590119909119909

120597119909

+

120597120590119909119911

120597119911

120588

1205972V

1205971199052=

120597120590119911119909

120597119909

+

120597120590119911119911

120597119911

(9)


120590119911119911= 120590119911119909= 0

120597119899

120597119911

= 0 for 119911 = 0


(10)


1198991(1198990



1205971198991

120597119905

+

1198991

120591

minus 119863(

12059721198991

1205971199092+

12059721198991

1205971199112)

= minus

11989901205730119863

119896119861119879

[

1205972119890

1205971199092+

1205972119890

1205971199112+ 1198922(

1205974119890

1205971199094+

1205972119890

12059711990921205971199112)]

(11)


1) and 120597119899

1120597119905 in




1198991 (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

119899 (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

119906 V (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

V (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

(12)


120597119911119909

120597119911

minus 119894119902119909119909+ 1205881205962 = 0

120597119911119911

120597119911

minus 119894119902119909119911+ 1205881205962V = 0

1205972119899

1205971199112minus (1199022+

119894120596120591 + 1

1198972

) 119899

=

1198990120573

119896119861119879

(minus1198941199023 + 1199022 1205972V

1205971199112+ 119894119902

1205972

1205971199112minus

1205973V

1205971199113)

(13)

where

119909119909

= minus119894119902 (120582 + 2120583) + 120582

120597V

120597119911

minus 120573119899

119911119911= (120582 + 2120583)

120597V

120597119911

minus 119894119902120582 minus 120573119899

119909119911= 120583(

120597

120597119911

minus 119894119902V)

(14)


119903119902


119903= Re(120596) and


of 120596 120582 = 120582(119902) = 1205820(1 minus 119892

21199022) and 120583 = 120583(119902) = 120583

0(1 minus 119892

21199022)


21199022) is



= 1198861exp (minus120578

1119911) + 119886

2exp (minus120578

2119911) + 119886

3exp (minus120578

3119911)

V = minus

119894119902

1205781

1198861exp (minus120578

1119911) + 119898

21198862exp (minus120578

2119911)

+ 11989831198863exp (minus120578

3119911)

119899 = 11988921198862exp (minus120578

2119911) + 119889

31198863exp (minus120578

3119911)

(15)

where

1205782

1= 1199022minus

1205881205962

120583

(16)

1205782



1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]

+(1199022 minus 120588 1205962

120582 + 2120583

)(

119894120596120591 + 1

1198972

+ 1199022) minus 120575119902

4 = 0

(17)


01205732(120582 +


interaction and

119898119895=

120578119895

119894119902

119889119895=

120582 + 2120583

120573119894119902

(1205782

1minus 1205782

119895) 119895 = 2 3 (18)



1205782

2+ 1205782

3= [2119902

2(1 minus 120575) minus 120588

1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]


1205782

21205782

3= [(119902

2minus 120588

1205962

120582 + 2120583

)(1199022+

119894120596120591 + 1

1198972

) minus 1205751199024]


(19)


119909119909(119902 119911) = minus 2119894119902 (120582 + 120583) 119886

1exp (minus120578

1119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057821198982+ 1205731198892) 1198862exp (minus120578

2119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057831198983+ 1205731198893) 1198863exp (minus120578

3119911)

119911119911(119902 119911) = 2119894119902120583119886

1exp (minus120578

1119911)

+ (119894119902120582 minus (120582 + 2120583) 12057821198982+ 1205731198892) 1198862exp (minus120578

2119911)


3119911)

119909119911(119902 119911) = 120578

1(1 +

1199022

1205782

1

)1198861exp (minus120578

1119911)

+ (1205782+ 119894119902119898

2) 1198862exp (minus120578

2119911)

+ (1205783+ 119894119902119898

3) 1198863exp (minus120578

3119911)

(20)



119895= 0 119895 = 1 2 3 in (16) and (17) we obtain

1205881205962= 120583 (119902) 119902

2 (21)

(1199022+

119894120596120591 + 1

1198972

)[1199022minus 1205962 120588

120582 (119902) + 2120583 (119902)

] minus 120575 (119902) 1199024= 0

(22)


0= const

120583(119902) = 1205830= const and 120575(119902) = 120575



1205962= 1198882

11987901199022 (23a)

(1199022minus 1205962119888minus2

1198710) [1199022+ (119894120596120591 + 1) 119897

minus2] minus 12057501199024= 0 (23b)

(1198881198710

= [(21205830+ 1205820)120588]12 and 119888

1198790= (120583

0120588)12 are the




12= plusmn119888119871119902




12059612

= 120596(0)

12+ 120575120596(1)

12+ 1205752120596(2)

12+ sdot sdot sdot

1205963= 120596(0)

3+ 120575120596(1)

3+ 1205752120596(2)

3+ sdot sdot sdot

(24)


12059612

= 120596(0)

12minus

120575

2

120596(0)

1211990221198972(11990221198972+ 1 minus 119894120596

(0)

12120591)

(11990221198972+ 1)2+ (120596(0)

12120591)

2+ 119874 (120575

2) (25)

1205963= 120596(0)

3minus

11989412057511989721198882

1198711199024120591

1198882

11987111990221205912+ (11990221198972+ 1)2+ 119874 (120575

2) (26)

From (25) we obtain

Re (12059612) = 120596(0)

12[1 minus

120575

2

11990221198972(11990221198972+ 1)

(11990221198972+ 1)2+ (119888119871119902120591)2] (27)

Im (12059612) =

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2+ (119888119871119902120591)2 (28)

If 119888119871119902 ≫ 120591



and 120578

119871nabla(div

)where 120578

119879= 120578 120578



Im (12059612) =

1

2120588

1205781199022+

120575

2120591

(11989721199022+ 1) (29)


Re (12059612) = 120596(0)

12[1 minus

120575

2120591

(11989721199022+ 1)

119888119871119902

] asymp 120596(0)

12 (30)



If 119888119871119902 ≪ 120591

minus1(11989721199022+ 1)

Re (12059612) = 120596(0)

12(1 minus

120575 11990221198972

2 (11990221198972+ 1)

) (31)

Im (12059612) =

1

2120588

1205781199022+

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2 (32)


12) rarr 0 Im(120596

12) gt 0)



120578=

12057811990222120588 ≪ 119888




Γ120591 = 11989721199022(120575 minus 1) minus 1 (33)

or

Γ = 1198631199022[

[

1205750

(1 minus 11989221199022)

2

1 minus ℎ21199022

minus 1]

]

minus

1

120591

(34)

where 1205750= 11989901205732

0(1205820+ 21205830)119896119861119879



120591 = 1205911198631198922 and parameters 120574 = 119892



Γ =

1199022

1 minus 1199022(1199022+ 1205750(1 minus 120574119902

2)

2

minus 1) minus

1

120591

(35)




119898=







Γ (119902) = 1198631199022[

119866

119866119888


1

120591

(36)

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)



0= 10

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)





where

119866119888= 119896119861119879120588 1198882

119871(1205911205732

0)

minus1

(37)


0=

20 eV 1205881198882119871= 1012 erg cmminus3 119879 = 10





at 119902 =


119898of the


119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)


Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888











0) of


119898119896119861119879) into (37) for the


119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)




119888) from (38)





211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)


(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578




1 1205782 and 120578

3



4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)



(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783




119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120590119911119911= (1 + 119892

2nabla2) [(1205820+ 21205830)

120597V

120597119911

+ 1205820

120597119906

120597119909

]

minus (1 + ℎ2nabla2) 1205730119899

120590119909119911= (1 + 119892

2nabla2) 1205830(

120597119906

120597119911

+

120597V

120597119909

)

(8)


120588

1205972119906

1205971199052=

120597120590119909119909

120597119909

+

120597120590119909119911

120597119911

120588

1205972V

1205971199052=

120597120590119911119909

120597119909

+

120597120590119911119911

120597119911

(9)


120590119911119911= 120590119911119909= 0

120597119899

120597119911

= 0 for 119911 = 0


(10)


1198991(1198990



1205971198991

120597119905

+

1198991

120591

minus 119863(

12059721198991

1205971199092+

12059721198991

1205971199112)

= minus

11989901205730119863

119896119861119879

[

1205972119890

1205971199092+

1205972119890

1205971199112+ 1198922(

1205974119890

1205971199094+

1205972119890

12059711990921205971199112)]

(11)


1) and 120597119899

1120597119905 in




1198991 (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

119899 (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

119906 V (119909 119911 119905)

=

1

2120587

∬

infin

minusinfin

V (119902 119911 120596) exp [minus119894 (119902119909 + 120596119905)] 119889119902 119889120596

(12)


120597119911119909

120597119911

minus 119894119902119909119909+ 1205881205962 = 0

120597119911119911

120597119911

minus 119894119902119909119911+ 1205881205962V = 0

1205972119899

1205971199112minus (1199022+

119894120596120591 + 1

1198972

) 119899

=

1198990120573

119896119861119879

(minus1198941199023 + 1199022 1205972V

1205971199112+ 119894119902

1205972

1205971199112minus

1205973V

1205971199113)

(13)

where

119909119909

= minus119894119902 (120582 + 2120583) + 120582

120597V

120597119911

minus 120573119899

119911119911= (120582 + 2120583)

120597V

120597119911

minus 119894119902120582 minus 120573119899

119909119911= 120583(

120597

120597119911

minus 119894119902V)

(14)


119903119902


119903= Re(120596) and


of 120596 120582 = 120582(119902) = 1205820(1 minus 119892

21199022) and 120583 = 120583(119902) = 120583

0(1 minus 119892

21199022)


21199022) is



= 1198861exp (minus120578

1119911) + 119886

2exp (minus120578

2119911) + 119886

3exp (minus120578

3119911)

V = minus

119894119902

1205781

1198861exp (minus120578

1119911) + 119898

21198862exp (minus120578

2119911)

+ 11989831198863exp (minus120578

3119911)

119899 = 11988921198862exp (minus120578

2119911) + 119889

31198863exp (minus120578

3119911)

(15)

where

1205782

1= 1199022minus

1205881205962

120583

(16)

1205782



1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]

+(1199022 minus 120588 1205962

120582 + 2120583

)(

119894120596120591 + 1

1198972

+ 1199022) minus 120575119902

4 = 0

(17)


01205732(120582 +


interaction and

119898119895=

120578119895

119894119902

119889119895=

120582 + 2120583

120573119894119902

(1205782

1minus 1205782

119895) 119895 = 2 3 (18)



1205782

2+ 1205782

3= [2119902

2(1 minus 120575) minus 120588

1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]


1205782

21205782

3= [(119902

2minus 120588

1205962

120582 + 2120583

)(1199022+

119894120596120591 + 1

1198972

) minus 1205751199024]


(19)


119909119909(119902 119911) = minus 2119894119902 (120582 + 120583) 119886

1exp (minus120578

1119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057821198982+ 1205731198892) 1198862exp (minus120578

2119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057831198983+ 1205731198893) 1198863exp (minus120578

3119911)

119911119911(119902 119911) = 2119894119902120583119886

1exp (minus120578

1119911)

+ (119894119902120582 minus (120582 + 2120583) 12057821198982+ 1205731198892) 1198862exp (minus120578

2119911)


3119911)

119909119911(119902 119911) = 120578

1(1 +

1199022

1205782

1

)1198861exp (minus120578

1119911)

+ (1205782+ 119894119902119898

2) 1198862exp (minus120578

2119911)

+ (1205783+ 119894119902119898

3) 1198863exp (minus120578

3119911)

(20)



119895= 0 119895 = 1 2 3 in (16) and (17) we obtain

1205881205962= 120583 (119902) 119902

2 (21)

(1199022+

119894120596120591 + 1

1198972

)[1199022minus 1205962 120588

120582 (119902) + 2120583 (119902)

] minus 120575 (119902) 1199024= 0

(22)


0= const

120583(119902) = 1205830= const and 120575(119902) = 120575



1205962= 1198882

11987901199022 (23a)

(1199022minus 1205962119888minus2

1198710) [1199022+ (119894120596120591 + 1) 119897

minus2] minus 12057501199024= 0 (23b)

(1198881198710

= [(21205830+ 1205820)120588]12 and 119888

1198790= (120583

0120588)12 are the




12= plusmn119888119871119902




12059612

= 120596(0)

12+ 120575120596(1)

12+ 1205752120596(2)

12+ sdot sdot sdot

1205963= 120596(0)

3+ 120575120596(1)

3+ 1205752120596(2)

3+ sdot sdot sdot

(24)


12059612

= 120596(0)

12minus

120575

2

120596(0)

1211990221198972(11990221198972+ 1 minus 119894120596

(0)

12120591)

(11990221198972+ 1)2+ (120596(0)

12120591)

2+ 119874 (120575

2) (25)

1205963= 120596(0)

3minus

11989412057511989721198882

1198711199024120591

1198882

11987111990221205912+ (11990221198972+ 1)2+ 119874 (120575

2) (26)

From (25) we obtain

Re (12059612) = 120596(0)

12[1 minus

120575

2

11990221198972(11990221198972+ 1)

(11990221198972+ 1)2+ (119888119871119902120591)2] (27)

Im (12059612) =

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2+ (119888119871119902120591)2 (28)

If 119888119871119902 ≫ 120591



and 120578

119871nabla(div

)where 120578

119879= 120578 120578



Im (12059612) =

1

2120588

1205781199022+

120575

2120591

(11989721199022+ 1) (29)


Re (12059612) = 120596(0)

12[1 minus

120575

2120591

(11989721199022+ 1)

119888119871119902

] asymp 120596(0)

12 (30)



If 119888119871119902 ≪ 120591

minus1(11989721199022+ 1)

Re (12059612) = 120596(0)

12(1 minus

120575 11990221198972

2 (11990221198972+ 1)

) (31)

Im (12059612) =

1

2120588

1205781199022+

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2 (32)


12) rarr 0 Im(120596

12) gt 0)



120578=

12057811990222120588 ≪ 119888




Γ120591 = 11989721199022(120575 minus 1) minus 1 (33)

or

Γ = 1198631199022[

[

1205750

(1 minus 11989221199022)

2

1 minus ℎ21199022

minus 1]

]

minus

1

120591

(34)

where 1205750= 11989901205732

0(1205820+ 21205830)119896119861119879



120591 = 1205911198631198922 and parameters 120574 = 119892



Γ =

1199022

1 minus 1199022(1199022+ 1205750(1 minus 120574119902

2)

2

minus 1) minus

1

120591

(35)




119898=







Γ (119902) = 1198631199022[

119866

119866119888


1

120591

(36)

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)



0= 10

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)





where

119866119888= 119896119861119879120588 1198882

119871(1205911205732

0)

minus1

(37)


0=

20 eV 1205881198882119871= 1012 erg cmminus3 119879 = 10





at 119902 =


119898of the


119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)


Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888











0) of


119898119896119861119879) into (37) for the


119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)




119888) from (38)





211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)


(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578




1 1205782 and 120578

3



4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)



(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783




119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1205782

2+ 1205782

3= [2119902

2(1 minus 120575) minus 120588

1205962

120582 + 2120583

+

119894120596120591 + 1

1198972

]


1205782

21205782

3= [(119902

2minus 120588

1205962

120582 + 2120583

)(1199022+

119894120596120591 + 1

1198972

) minus 1205751199024]


(19)


119909119909(119902 119911) = minus 2119894119902 (120582 + 120583) 119886

1exp (minus120578

1119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057821198982+ 1205731198892) 1198862exp (minus120578

2119911)

+ (minus119894119902 (120582 + 2120583) + 12058212057831198983+ 1205731198893) 1198863exp (minus120578

3119911)

119911119911(119902 119911) = 2119894119902120583119886

1exp (minus120578

1119911)

+ (119894119902120582 minus (120582 + 2120583) 12057821198982+ 1205731198892) 1198862exp (minus120578

2119911)


3119911)

119909119911(119902 119911) = 120578

1(1 +

1199022

1205782

1

)1198861exp (minus120578

1119911)

+ (1205782+ 119894119902119898

2) 1198862exp (minus120578

2119911)

+ (1205783+ 119894119902119898

3) 1198863exp (minus120578

3119911)

(20)



119895= 0 119895 = 1 2 3 in (16) and (17) we obtain

1205881205962= 120583 (119902) 119902

2 (21)

(1199022+

119894120596120591 + 1

1198972

)[1199022minus 1205962 120588

120582 (119902) + 2120583 (119902)

] minus 120575 (119902) 1199024= 0

(22)


0= const

120583(119902) = 1205830= const and 120575(119902) = 120575



1205962= 1198882

11987901199022 (23a)

(1199022minus 1205962119888minus2

1198710) [1199022+ (119894120596120591 + 1) 119897

minus2] minus 12057501199024= 0 (23b)

(1198881198710

= [(21205830+ 1205820)120588]12 and 119888

1198790= (120583

0120588)12 are the




12= plusmn119888119871119902




12059612

= 120596(0)

12+ 120575120596(1)

12+ 1205752120596(2)

12+ sdot sdot sdot

1205963= 120596(0)

3+ 120575120596(1)

3+ 1205752120596(2)

3+ sdot sdot sdot

(24)


12059612

= 120596(0)

12minus

120575

2

120596(0)

1211990221198972(11990221198972+ 1 minus 119894120596

(0)

12120591)

(11990221198972+ 1)2+ (120596(0)

12120591)

2+ 119874 (120575

2) (25)

1205963= 120596(0)

3minus

11989412057511989721198882

1198711199024120591

1198882

11987111990221205912+ (11990221198972+ 1)2+ 119874 (120575

2) (26)

From (25) we obtain

Re (12059612) = 120596(0)

12[1 minus

120575

2

11990221198972(11990221198972+ 1)

(11990221198972+ 1)2+ (119888119871119902120591)2] (27)

Im (12059612) =

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2+ (119888119871119902120591)2 (28)

If 119888119871119902 ≫ 120591



and 120578

119871nabla(div

)where 120578

119879= 120578 120578



Im (12059612) =

1

2120588

1205781199022+

120575

2120591

(11989721199022+ 1) (29)


Re (12059612) = 120596(0)

12[1 minus

120575

2120591

(11989721199022+ 1)

119888119871119902

] asymp 120596(0)

12 (30)



If 119888119871119902 ≪ 120591

minus1(11989721199022+ 1)

Re (12059612) = 120596(0)

12(1 minus

120575 11990221198972

2 (11990221198972+ 1)

) (31)

Im (12059612) =

1

2120588

1205781199022+

120575120591

2

1198882

11987111990241198972

(11990221198972+ 1)2 (32)


12) rarr 0 Im(120596

12) gt 0)



120578=

12057811990222120588 ≪ 119888




Γ120591 = 11989721199022(120575 minus 1) minus 1 (33)

or

Γ = 1198631199022[

[

1205750

(1 minus 11989221199022)

2

1 minus ℎ21199022

minus 1]

]

minus

1

120591

(34)

where 1205750= 11989901205732

0(1205820+ 21205830)119896119861119879



120591 = 1205911198631198922 and parameters 120574 = 119892



Γ =

1199022

1 minus 1199022(1199022+ 1205750(1 minus 120574119902

2)

2

minus 1) minus

1

120591

(35)




119898=







Γ (119902) = 1198631199022[

119866

119866119888


1

120591

(36)

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)



0= 10

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)





where

119866119888= 119896119861119879120588 1198882

119871(1205911205732

0)

minus1

(37)


0=

20 eV 1205881198882119871= 1012 erg cmminus3 119879 = 10





at 119902 =


119898of the


119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)


Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888











0) of


119898119896119861119879) into (37) for the


119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)




119888) from (38)





211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)


(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578




1 1205782 and 120578

3



4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)



(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783




119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120578=

12057811990222120588 ≪ 119888




Γ120591 = 11989721199022(120575 minus 1) minus 1 (33)

or

Γ = 1198631199022[

[

1205750

(1 minus 11989221199022)

2

1 minus ℎ21199022

minus 1]

]

minus

1

120591

(34)

where 1205750= 11989901205732

0(1205820+ 21205830)119896119861119879



120591 = 1205911198631198922 and parameters 120574 = 119892



Γ =

1199022

1 minus 1199022(1199022+ 1205750(1 minus 120574119902

2)

2

minus 1) minus

1

120591

(35)




119898=







Γ (119902) = 1198631199022[

119866

119866119888


1

120591

(36)

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)



0= 10

02

01

minus01

minus02

0

0

01 02 03 04 05

Non

dim

ensio

nal g

row

th ra

te (Γ

)





where

119866119888= 119896119861119879120588 1198882

119871(1205911205732

0)

minus1

(37)


0=

20 eV 1205881198882119871= 1012 erg cmminus3 119879 = 10





at 119902 =


119898of the


119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)


Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888











0) of


119898119896119861119879) into (37) for the


119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)




119888) from (38)





211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)


(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578




1 1205782 and 120578

3



4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)



(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783




119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





at 119902 =


119898of the


119889latt = radic8120587119892radic

119866

119866 minus 119866119888

(38)


Γ119898=

119863

41198922

(1 minus 119866119888119866)2

119866119866119888











0) of


119898119896119861119879) into (37) for the


119866119888 (119879) =

120588 1198882

119871119896119861119879

12059101205730

2exp(minus

119908119898

119896119861119879

) (40)




119888) from (38)





211990221198861+ (1205782

1+ 1199022) 1198862minus (1205782

1+ 1199022) 1198863= 0

(1205782

1+ 1199022) 1198861+ 2120578212057811198862+ 2120578312057811198863= 0

1205782(1205782

2minus 1199022+

1205881205962

120582 + 2120583

)1198862+ 1205783(1205782

3minus 1199022+

1205881205962

120582 + 2120583

)1198863= 0

(41)


(1205782

1+ 1199022)

2

=

41205781120578212057831199022(1205782+ 1205783)

1205782

2+ 1205782

3+ 12057821205783+ 1205881205962(120582 + 2120583)

minus1minus 1199022

(42)

where 1205781 1205782 and 120578




1 1205782 and 120578

3



4119902212057811205782minus (1205782

1+ 1199022)

2

= 0 (43)

1205782

1= 1199022minus

1205881205962

120583

1205782

2= 1199022minus

1205881205962

(120582 + 2120583)

(44)



(1 +

1205782

1

1199022)

2

=

41205781119861(119860 + 2119861)

12

1199022(119860 + 119861 minus 120578

2

2)

(45)

where 119860 = 1205782

2+ 1205782

3and 119861 = 120578

21205783




119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119861 = 12057821205783(1 + 120575

(1205782

21205782

3minus 1199024)

21205782

21205782

3

)

119860 = 1205782

2+ 1205782

3+ 120575 (120578

2

2+ 1205782

3minus 21199022) 120578

2

3= 1199022+

(119894120596120591 + 1)

1198972

(46)


119877 (120596 119902) minus 120575119885 (120596 119902) = 0 (47)

Here

119877 (120596 119902) = (1205782

1+ 1199022)

2

minus 4119902212057811205782

119885 (120596 119902) =

211990221205781(21205782+ 1205783) (1205782

2minus 1199022)

2

12057831205782(1205782+ 1205783)2

(48)



following form

119877 (120585) minus 120575 119885 (120585) = 0 (49)


119877 (120585119877+ 120576) minus 119885 (120585

119877+ 120576) 120575 = 0 (50)


119877) = 0


119877


120576 (120585119877 120575) = minus

119885 (120585119877)

(120597119877120597120585)1003816100381610038161003816120585=120585119877

120575 = 120576119903+ 119894120576119894 (51)

where 120576119903= Re(120576) and 120576

119894= Im(120576)



constant We have

120585 =

1

1198882

1

(

120596

119902

)

2

= 120585119877+ 120576 120596 = 119902119888

1(120576119903+ 120585119877+ 119894120576119894)12 (52)


120596119903= 1198881119902radic120585119877(

1 minus 1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (53a)

120596119894=

1198881119902120575119885119894

120585119877119872

(53b)

where 119885119903= Re(Ζ) 119885

119894= Im(Ζ) and119872 = (120597119877120597120585)|

120585=120585119877


119888 = 1198881radic120585119877(1 minus

1205751003816100381610038161003816119885119903

1003816100381610038161003816

2120585119877119872

) (54)


119903rarr 0 and120596

119894gt 0)




120596119894119888119879sdot119871




Γ = 1198631199022[

[

120575


(1 minus ℎ21199022)

2

(1 minus 11989221199022)

minus 1]

]


where 120577 = 1198882

119871119888minus2

119879






Γ =

1199022

1 minus 1199022(1199022+ 120601(1 minus 120574119902

2)

2






119898




Γ = 1199022[120601 (1 minus 119902


minus1 (57)


119902 = 119902119898=

1

radic2

radic1 minus

119866(119904)

119888

119866

(58)

where

119866 gt 119866(119904)

119888=

119896119861119879120588 1198882

119871(1 minus 119888

2

119879119888minus2

119871)

1205911205732

0

(59)





119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119898is given by

119889(119904)

latt =2120587

119902119898


119866

119866 minus 119866(119904)

119888

(60)


Γ119898=

119863

41198922

119866

119866(119904)

119888

(1 minus

119866

119866(119904)

119888

)

2










1198990= 1198660120591 = 119866

01205910exp(

minus1199081198890

1198961198611198790

) (62)

where

1199081198890= 1199081198910minus 1199081198980 119866

0= 119866 (119879

0) (63)



0are


119879= (120594120591Las)


1198790=

2 (1 minus 119877) 1198680

120582119879

radic

120594120591Las120587

(64)



119879=


0= 12times10


0= 12times10


119889= 10minus22 cm3 119870 = 510

11 ergcm3 120577 = 244and119908



119888120591) which


0=

30 eV 119888119871

= 24 times 105 cms 119866 = 11119866

119888 2120587119892 = 15 nm



119898= 6 times 10


119898120591Las = 12 ≫ 1)




7 Conclusions







References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








References



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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