Mathematical Physics in Diffusion Problems

Transcription

1 Joural of Moder Physics Published Olie ovember 5 i SciRes. Mathematical Physics i Diffusio Problems Takahisa Okio Departmet of Applied Mathematics Faculty of Egieerig Oita Uiversity Oita ity Japa Received 4 October 5; accepted 4 ovember 5; published 7 ovember 5 opyright 5 by author ad Scietific Research Publishig c. This work is licesed uder the reative ommos Attributio teratioal Licese ( BY). Abstract Usig the divergece theorem ad the coordiate trasformatio theory for the geeral Fickia secod law fudametal diffusio problems are ivestigated. As a result the ew fidigs are obtaied as follows. The uified diffusio theory is reasoably established icludig a self-diffusio theory ad a ( ) elemets system iterdiffusio oe. The Fickia first law is icomplete without a costat diffusio flux correspodig to the Brow motio i the localized space. The cause of Kirkedall effect ad the oexistece of itrisic diffusio cocept are theoretically revealed. the parabolic space a elegat aalytical method of the diffusio equatio is mathematically established icludig a oliear diffusio equatio. From the Schrödiger equatio ad the diffusio equatio the uiversal expressio of diffusivity proportioal to the Plack costat is reasoably obtaied. The material wave equatio proposed by de Broglie is also derived i relatio to the Brow motio. The fudametal diffusio theories discussed here will be highly useful as a stadard theory for the basic study of actual iterdiffusio problems such as a alloy a compoud semicoductor a multilayer thi film ad a microstructure material. Keywords Brow Particle Boltzma Factor Markov Process Parabolic Law Error Fuctio. troductio First of all we state that the basic diffusio equatio of the geeral oliear Fickia secod law is discussed i accordace with the fudametal mathematical physics i the preset work. The exteded diffusio equatios i detail are ot thus discussed. evertheless the ew fidigs which are extremely domiat i the diffusio study are reasoably obtaied. the diffusio history the problems relevat to the coordiate trasformatio of diffusio equatio had ot bee discussed i accordace with the Gauss divergece theorem util recetly. That is ust a reaso why the ew diffusio theories are discussed i the preset study. t will be gradually clarified i the text that the coordiate trasformatio theory is essetially idispesable for the diffusio study. t is obvious that aalyzig the exteded diffusio equatio must be based o the fudametal diffusio theory. The How to cite this paper: Okio T. (5) Mathematical Physics i Diffusio Problems. Joural of Moder Physics

2 ew fudametal fidigs differet from the existig diffusio theories obtaied here will thus exert a great ifluece o the actual diffusio problems i detail ust because of fudametal oes. A great may pheomea i various sciece fields are expressed by usig the well-kow evolutio equatios. The diffusio equatio is oe of them ad mathematically correspods to the Markov process i relatio to the ormal distributio rule []. other words the motio of diffusio particles correspods to the well-kow Brow movemet satisfyig the parabolic law [] [3]. t is widely accepted that the Brow problem is a geeral term of ivestigatig subects i various sciece fields relevat to the Markov process such as material sciece iformatio sciece life sciece ad social sciece [4]-[9]. physics we ca also uderstad the diffusio equatio i accordace with the Gauss divergece theorem []. f we apply the divergece theorem to the diffusio problem for a material uder the coditio of o sik ad source of the material it is foud that the material coservatio law is valid for the diffusio particles regardless of a thermodyamic state of material. The diffusio equatio is also called the cotiuous equatio ad is extremely fudametal oe i physics. history the heat coductio equatio which is mathematically equivalet to the diffusio equatio was proposed by Fourier regardless of the Markov process ad the divergece theorem []. accordace with the idustrial requiremet the solid materials such as alloys semicoductors ad multilayer materials have bee widely fabricated. The heat treatmet is idispesable for their fabricatio processes the. The migratio of particles i a material is caused by the heat treatmet. relatio to the migratio of their particles the diffusio problems of various solid materials have bee thus widely ivestigated []. Therefore the diffusio problem is a fudametal study subect i the materials sciece icludig the cases of liquid ad gas states. the preset work the fudametal problems of the geeral Fickia secod law where a drivig force affects the diffusio system are discussed i accordace with the mathematical theory. The preset aalytical method is applicable to iterdiffusio problems of a elemets system of every material i a arbitrary thermodyamic state. Although the physical validity of the preset method is ivestigated by usig the diffusio data cocerig the solid metals the mathematical geerality discussed here is still kept. The heat coductio equatio proposed by Fourier i 8 has bee applied to ivestigatig the temperature distributio i materials []. 87 the so-called Brow motio was foud where the self-diffusio of water was visualized by polle micro particle motios [] [3]. 855 Fick applied the heat coductio equatio to diffusio pheomea as it had bee [3]. evertheless the Brow motio had ot bee recogized as a diffusio problem util the Eistei theory of Brow motio i 95 although it was a typical diffusio problem [3]. Although the cocetratio of diffusio particles is a real quatity i physics the temperature is a thermodyamic state quatity. As far as the shape of heat coductio material is uchageable durig a thermal treatmet the coordiate system of heat coductio equatio set i a material is a fixed oe sice the coordiate system is ot iflueced by variatios of the material iteral structure. O the other had strictly speakig the coordiate system of diffusio equatio set i the diffusio field (solvet) is a movig oe sice it is geerally iflueced by such variatios. Whe the Fickia first ad secod laws (F law ad F law) were proposed the Gauss divergece theorem had bee already reported i 84 []. evertheless the problem of coordiate system of diffusio equatio was ot mathematically ivestigated i accordace with the divergece theorem. geeral it is idispesable for uderstadig the diffusio problems to discuss their coordiate systems sice it is strictly speakig cosidered that the diffusio particles solvet particles ad also the diffusio regio space simultaeously move agaist the experimetatio system i the diffusio regio outside. Although the Fickia laws were still widely applied to various diffusio pheomea as essetial equatios i physics the problem betwee coordiate systems was ot discussed. Recetly the diffusio equatio was thus mathematically ivestigated i accordace with the divergece theorem ad the coordiate trasformatio theory [4]-[6]. t is revealed i the text that the diffusio flux should be determied by takig accout of the cocered coordiate system of diffusio equatio. Usig the correspodig diffusio flux to the coordiate system of diffusio equatio for iterdiffusio oe way diffusio impurity diffusio ad self-diffusio they are uiformly discussed i the text. As a result we foud that the foudatio of diffusio problems is icluded i iterdiffusio problems. The iterdiffusio theory of a ( ) elemets system applicable to every material was thus reasoably established [6]. the aalysis of iterdiffusio problems the oly differece betwee a biary system ad a elemets system is

3 whether the solvet material is oe elemet or ( ) elemets. The coordiate trasformatio theory reveals that the correspodig F law to the F law is icomplete without a costat diffusio flux relevat to the Brow motio i the localized space. Further it was also foud that the Kirkedall effect (K effect) is caused by a shift betwee the coordiate systems of the diffusio equatio like the Doppler effect relevat to a wave equatio is caused by a shift betwee the fixed coordiate system ad the movig oe for the wave equatio [4]-[7]. At the same time this meas that the cocept of itrisic diffusio is uecessary for uderstadig the K effect. As far as the sik ad source of a material is oexistet i the diffusio system the Gauss divergece theory shows that the geeral F law satisfies the material coservatio law eve if a drivig force affects the diffusio system. The drivig force affects etropy ad ump frequecies of diffusio particles i the material. The ifluece of a drivig force is thus all icorporated ito the diffusivity of diffusio equatio the. the existig diffusio theory the diffusio fluxes iflueced by a drivig force are discussed icorporatig ot ifluece of both etropy ad ump frequecies but oly that of ump frequecies o the diffusivity ito them for example such as a drift velocity. However such cocept as a drift velocity is ot ecessary for aalyzig the diffusio equatio sice we ca solve a oliear diffusio equatio i accordace with the fudametal theory i the mathematical physics regardless of the discussio about diffusio fluxes. The diffusivity is obtaied from the Taylor expasio of a probability desity fuctio satisfyig the parabolic law [3]. This idicates that the aalysis of diffusio equatio is essetially possible i the parabolic space. The diffusivity is defied by a iteractio betwee a diffusio particle ad the diffusio field ear the diffusio particle itself. This idicates that the diffusivity should be essetially ivestigated i the quatum mechaics sice the behavior of a micro particle should be ivestigated by aalyzig the Schrödiger Equatio (S equatio) [8]. Based o the suggestios metioed above the elegat aalytical method i the parabolic space is reasoably established i the text [9] []. As a result the solutios of the Boltzma trasformatio equatio for oliear iterdiffusio pheomea were for the first time obtaied as aalytical expressios i the parabolic space []. From applyig the diffusio equatio to a problem of diffusio elemetary process we derived the S equatio []. t is revealed i the text that the diffusivity correspods to the agular mometum operator i the quatum mechaics. As a result the uiversal expressio of diffusivity which is applicable to every material i a arbitrary thermodyamic state was obtaied i proportioal to the Plack costat. t was also foud that the well-kow material wave relatio proposed by de Broglie i 93 which is the most fudametal oe i materials sciece is obtaied from a relatio betwee diffusivity expressios [3]. This gives evidece for the theory discussed i the preset study. the preset work we review the fudametal diffusio theories relevat to the geeral F law where they are systematically reframed i poits of view differet from the previous works addig some ew discussios to them. The ew fidigs obtaied here will be widely applicable to fudametal problems as a stadard theory i various actual diffusio pheomea.. Fudametal Theory of Diffusio Equatio As far as a material is coserved i the give regio the divergece theory shows that the diffusio equatio is applicable to diffusio pheomea of every material i a arbitrary thermodyamic state. The diffusio iformatio of a material such as crystal material or amorphous material ad/or solid liquid ad gas states is all icorporated ito the diffusivity i the give diffusio equatio. The diffusio equatio with such a arbitrary diffusivity is thus ivestigated i the followig. the preset study a abbreviate differetial otatio for a arbitrary idepedet variable ad the wellkow Dirac s bracket otatio for a arbitrary vector are used as follows [4]; = = ( x y z). f a operator Q is Hermite oe Q { Q } thus defied as = { } because of { } = is valid i the Hermite cougate. Here the otatio is =.

4 .. Diffusio Equatio t r is defied as a ormalized cocetratio where a diffusio particle i the iitial state The fuctio ( ) ( t r ) exists i the time ad space state ( ) t r after times umps. A diffusio particle moves at radom. t is therefore cosidered that the ump frequecy t ad ump displacemet r = r r = r are equivalet i probability to their mea values of all diffusio particles i the collective system. t r t r i the isotropic space the relatio of Sice it is also cosidered that the probability of diffusio-ump from the state of ( ) to ( t r ) is equivalet to oe from the same state to ( ) ( t+ t r ) = { ( t r r ) + ( t r + r )} (-) is thus valid. The Taylor expasio of both sides of Equatio (-) yields ( ) ( ) t ( ) t+ t r = t r + t t r + (-) ( ) ( ) ( ) ( r) ( ) t r ± r = t r ± r t r + t r ±. (-3) The substitutio of Equatios (-) ad (-3) ito Equatio (-) gives Sice the averaged value D give by t ad ( r). t = (-4) t r are physically fiite it is cosidered that ( r) t becomes a costat D ( r) = satisfyig the well-kow parabolic law [3]. Substitutig Equatios (-5) ito Equatios (-4) the diffusio equatio is obtaied as t (-5) = D = D = D t (-6) where D is rewritte as D whe the diffusio-ump directio is ot isotropic resultig from the existece of a drivig force i the diffusio system. The existece of a drivig force affects diffusivity D ad the diffusivity D becomes a fuctio of the idepedet variables t ad r. that case Equatio (-6) gives the diffusio flux yieldig where df ( ) { ( )} J = D = D + Ddf t r (-7) D t r is a diffusivity caused by a drivig force. O the other had usig a drift velocity v caused by a drivig force F the approximate diffusio flux is expressed as F * J D v F = + (-8) i the usual textbooks (see Appedix A) [5]. However the diffusio flux JF = v F discussed here is differet from oe give by Jdf = Ddf. (-9) the preset study the discussio about diffusio fluxes is ot ecessary for aalyzig the diffusio equatio sice we ca solve the oliear diffusio equatio i accordace with the fudametal theory i the mathematical physics regardless of diffusio fluxes. the followig therefore Equatio (-6) is directly solved i accordace with the mathematical physics. t is

5 i the ature of thigs that its aalytical method is applicable to the diffusio problems relevat to a drivig force the. V xyz withi the closed For a collective system composed of ( = ) elemets i the regio ( ) surface S( xyz ) the divergece theorem yields where ( ) { t } = { } dxdd yz D dxdd yz (-) S S txyz ad D are a cocetratio ad a diffusivity for a elemet uder the coditio of o sik ad source of the elemet. The itegral calculatio of {} i Equatio (-) meas a iflow flux from the surface outside through the surface elemet dd yz ad y ad z are the accepted as costat values because of the multiple itegral calculatio. Therefore Equatio (-) yields the diffusio equatio of { } ( ) ( ) tx = D tx (-) t x x i the diffusio field. Defiig the outflow flux as a plus value the diffusio flux is mathematics Equatio (-) yields { } ( ) = ( ) d = ( ) d. (-) J tx tx x D tx x x t x x Jx ( t x) = JF ( t x) + Jx ( t) + Jeq for JF ( tx ) D x ( tx ) where JF ( tx ) is the F law ad Jx ( t) + Jeq is a itegral costat agaist x. physics Jx ( ) = (-3) t meas a movemet of a diffusio regio space caused by a movemet of the diffusio field. the coordiate system of diffusio regio iside it should be thus physically accepted as Jx ( t ) =. However Jx ( t) must be take ito accout i the coordiate system of diffusio regio outside. The itrisic diffusio flux J eq idepedet of the time ad space is relevat to the Brow motio i the localized space. t plays a importat role for uderstadig a self-diffusio mechaism... oordiate System of Diffusio Equatio As discussed later the uified theory of diffusio problems shows that the foudatio of diffusio is i the iterdiffusio problems. the iterdiffusio problems solvet particles as well as diffusio particles move i the diffusio regio space. Whe a micro particle i a material umps from a site M to a iterstice space if we call such a iterstice space micro hole icludig a vacacy i case of a crystal material the micro hole of site is aihilated ad a ew micro hole is geerated at the site M after umpig. The diffusio regio space iteracts with the space of diffusio system outside i accordace with the aihilatio ad/or geeratio of micro holes like the diffusio system becomes a thermal equilibrium state resultig from a atiomy betwee the priciple of icrease of etropy ad that of miimum of free eergy. The coordiate system of the basic diffusio equatio must be set i the diffusio field sice the diffusivity depeds o a iteractio betwee a diffusio particle ad the diffusio field ear the diffusio particle itself. geeral therefore the diffusio problem should be ivestigated amog three coordiate systems where the origis of two coordiate systems are a poit P of the coordiate system ( xyz ) set i the diffusio field (solvet) ad a poit Q of the coordiate system ( xyz ) set i the diffusio regio space composed of micro holes ad the third origi is a poit R of the coordiate system ( ηζ ) set i the diffusio regio outside. the preset work we ivestigate a iterdiffusio problem applicable to every material i a arbitrary thermodyamic state. t is thus ivestigated usig the origial diffusio equatio (-) of the coordiate system ( tx ) ad the diffusio equatios trasformed ito the coordiate systems of ( t x ) ad ( τ ) uder the coditio of t = t = τ. The origis P ( tx ) = ( ) ad Q ( t x ) = ( ) are set at a poit of solvet material ad at a poit of micro holes o the iitial iterface of a diffusio couple. The origi R ( τ ) = ( ) is set at a poit of the diffusio system outside. The x axis is perpedicular to the iterface the. The x axis ad also axis are parallel to the x axis uder the coditio of x= x = = at t = t = τ =. t x = v τ ad with t is geerally cosidered that P ( tx ) = ( ) ad Q ( ) ( ) move with a velocity RP ( ) a velocity vrq ( τ ) agaist R ( τ ) = ( ) respectively ad also that P ( tx ) = ( ) moves with a velocity 3

6 vqp ( t ) agaist Q ( t x ) = ( ) icludig a possibility of v ( ) v ( ) v ( t ) velocities of must be physically valid. RP τ R Q τ QP =. The relatio amog these vrp = vrq + vqp (-4) Hereafter we use the suffix P Q ad R for a physical quatity relevat to ( tx ) ( t x ) ad ( ) τ. Equatio (-3) is thus rewritte as ( ) = ( ) + for J ( tx) D ( tx) P F eq J tx J tx J Whe the coordiate origi of ( tx ) moves from that of ( t x) F =. (-3) x with a velocity v QP the relatios of t t = t x= x xsft for xsft = vqpdt (-5) are valid uder the coditio of x = x= at t = t =. The relatios of t x = + = + t t t t x t x betwee differetial operators are obtaied the. vqp t x = + = x x t x x x Usig Equatio (-6) for Equatio (-) the diffusio equatios of ( t x) where = ( t x ) ad D D ( t x) sice ( t x) ( x ) is obtaied as (-6) t = D v x QP (-7) =. Equatio (-7) leads to the diffusio flux of ( ) = ( ) + + ( ) J t x J t x v J Q F QP eq is i the diffusio regio iside. the same maer of the above the diffusio equatio of ( τ ) is obtaied as JF t x = D x (-8) ( ) = (-9) τ D vrp. Sice ( τ ) is i the diffusio regio outside the diffusio flux becomes J ( τ ) = J ( τ ) + v + J ( τ) + J ( ) where R ( ) ( τ ) = ( ). R F R P R eq J τ = D (-) F J τ is the diffusio flux of micro holes caused by the movemet of the diffusio field agaist 3. terdiffusio Problems The iterdiffusio problems of a elemets system i a diffusio couple composed of arbitrary materials A ad B are discussed as follows. a amorphous material we the estimate a specime cross sectio perpedicular to the x axis where each cross sectio iterval l correspods to the averaged ump distace i the solid state or to the averaged collisio distace i the gas or liquid state. a crystal material we estimate the ump distace l betwee the earest eighbor crystal cross sectios perpedicular to the x axis. Whe materials A ad B are i the solid state durig a thermal diffusio we coceive that the cross sectio S of the material A is uiform ad equal to that of the material B. f materials A ad B are i the fluid state we coceive that they are i the receptacle correspodig to the case of solid materials. The iterface of diffusio couple betwee materials A ad B is smoothly oied. that case the origi x = of coordiate system ( tx ) is set at a poit o the iitial iterface i the diffusio field ad the orietatio of x axis is defied as A B. The material A is composed of elemets ad we defie the ormalized cocetratio for = A 4

7 m i the iitial state also icludig a possibility of A = for = m. The ormalized cocetratio i the material B is similarly defied as B. The cocetratio ad diffusivity of a elemet are ad D at a diffusio time t i the diffusio re- gio xa x xb tx = B. elemets are cosidered to be a solvet also icludig a possibility where becomes a solvet itself i accordace with a give diffusio system. The cross sectio of diffusio system is uiform ad the shape variatio of diffusio couple is usually egligible durig the thermal diffusio. that case the relatio of ormalized cocetratios yieldig. The boudary values of cocetratio are ( tx A) = A ad ( B) The remaied ( ) = (3-) = is usually accepted o each cross sectio betwee xa x xb. Here Equatio (3-) is physically valid regardless of a coordiate system. 3.. orrelatio of Diffusio Fluxes with oordiate Systems Equatios (-7) ad (3-) yield because of = =. mathematics the partial differetial equatio (3-) of t ad x is rewritte as t t = = { } k x ( D x vqp ) = x ( D D ) x vqp = (3-) = = k k { } x QP ( ) ( ) Γ t = D D v = k usig a arbitrary fuctio Γ( t ) of t because of Γ ( t ) = depedet o t ad x. O the other had Γ ( t) + vqp x. Here ( ) D D meas a diffusio flux k x meas a diffusio flux oly depedet oly o t. k Therefore Equatio (3-) meas D D = for a arbitrary. The iterdiffusio coefficiet D is thus defied as (3-3) D= D = D = = D i the diffusio regio x A x x B. We ca easily cofirm that Equatio (3-3) does ot deped o a coordiate system. Their boudary values are thus obtaied as ( τ ) = D ad D ( ) Usig Equatio (3-3) for the F law the followig relatio of D A F ( τ) = = = is valid regardless of a coordiate system. The relatio of A τ = D. (3-4) B B J = D = D = (3-5) is also valid as discussed later i the self-diffusio theory. Usig Equatios (3-5) ad (3-6) for Equatio (-) the relatio of Jeq = (3-6) = ( τ ) ( τ) for JR ( τ) JR ( τ) J = J = v + J R R RP R = = (3-7) is obtaied. Here JR ( τ ) meas the total diffusio flux of micro holes agaist the origi of ( ) = τ. that sit- 5

8 uatio JR ( τ ) = vrq is valid i the preset coordiate system because of J ( ) v of the origi of ( t x ) agaist the origi of ( τ ) RQ R vrq. Equatio (3-7) is thus rewritte as τ < usig a velocity the same maer the followig relatios J = v v. (3-7) R R P RQ P P ( ) (3-8) = J = J tx = Q = Q ( ) = QP = J J t x v (3-9) are also obtaied from Equatios (-3) ad (-8). Equatios (3-7)-(3-9) show that the diffusio flux idepedet of the space coordiate correspods to the velocity betwee the coordiate systems. The relatio of JR = JQ + JP (3-) must be physically valid because of t = t = τ. Equatio (3-) the yields the previous equatio of the followig JQ ( vqp ) ( τ) ( τ) ( τ) v = v + v (-4) RP RQ QP because of Equatios (3-7)-(3-). = is ivestigated by usig the cocetratio differece = B A ad the diffusio uctio depth x u µ Dt γ betwee x= xa = gradiets the flux of diffusio field for a elemet k ( ad x = x B. Takig accout of their cocetratio k) is obtaied as J t D D ( ) = γ = γ { } k Q A B = k xu t = k (3-) where the suffix γ meas γ A if A B or γ B if A B ad µ = is tetatively adopted. k k Here J ( t ) is adopted istead of J ( t x ) as a approximate equatio because of J v ( t) Q Equatio (3-) yields Q J = v t = J t = D t ( k ) ( ) γ { } Q QP Q A B k= = Equatio (3-) shows the movemet of ( tx ) agaist ( t x ) = ( ). Substitutig Equatio (3-) ito Equatio (-4) yields because of t = t = τ. R P RQ γ A B τ = 3.. Kirkedall Effect i terdiffusio Problems the metallurgy field the iterface of ( ) ( ) ( t x ) = ( ). Q =. QP (3-) v = v + D { } (3-3) tx = is the so-called Matao iterface (M iterface) ad that of is the Kirkedall iterface (K iterface) [7] [6]. 947 Kirkedall foud i the biary system iterdiffusio experimetatio that the iert marker set o the K iterface i the iitial state moves from the M iterface durig a thermal diffusio ad also that the M iterface does ot move durig thermal diffusio i.e. v RP =. Sice the the displacemet betwee the M iterface ad the K iterface has bee called the Kirkedall effect (K effect). Usig a experimetal value m the K effect x is obtaied as eff m t regardless of the space coordiate ad it satisfies the parabolic law. eff x = (3-4) 6

9 As far as the material shape of a diffusio couple is uchageable the K effect also reveals that the total quatity of elemet diffusig across the K iterface is differet from that of elemet. other words for the H H micro hole cocetratio ad its thermal equilibrium value betwee xa x xb the relatio of H H H H < is valid betwee xa x if > is valid betwee x xb durig thermal diffu- H H sio ad vice versa. The value of becomes gradually zero i accordace with the thermodyamic priciple. the biary system iterdiffusio problems the F law shows that the relatio of ( ) J + J = D D = x is valid because of + = ad D = D = D is valid the for a iterdiffusio coefficiet D. O the other had it was also cosidered that the K effect is caused by a differece betwee the diffusio flux of elemet ad that of elemet across the K iterface. other words the so-called itrisic diffusio coefficiet satisfyig the relatio of Dit Dit was ewly coceived to uderstad the K effect the. However the defiitio of diffusivity shows that the diffusivity is proportioal to a statistical ump frequecy of a diffusio particle at a poit ( tx ) occupied by the diffusio particle itself. t is therefore physically difficult to accept what a diffusio particle of the elemet or has two diffusivities D ad D it or D ad D it at the same poit ( tx ) i the cocered diffusio field uder the coditio of D = D = D sice the statistical ump frequecy value of a diffusio particle should be physically oly oe i the cocered diffusio field. the preset diffusio theory the behavior of micro holes is visualized by the movemet of a iert marker i the diffusio regio because of the iert characteristic. Therefore the K effect meas a shift betwee the coordiate systems ( tx ) ad ( t x ) after the thermal diffusio. other words Equatio (3-) yields the displacemet betwee the K iterface ad the M iterface because of v RP = i accordace with the experimetal results. The K effect is thus obtaied as ( ) ( ) γ { } t eff Q P A B = x = v t dt = D t (3-5) satisfyig the parabolic law. The cocept of itrisic diffusio is thus uecessary for uderstadig the K effect. Further the K effect occurs i the iterdiffusio ot oly of metal crystal but also of every material sice it is caused by a shift betwee the coordiate systems. Equatio (3-5) gives evidece that the K effect is caused by a shift betwee the coordiate systems sice it does ot deped o the space coordiate. Substitutig = ito Equatio (3-5) ad comparig it with Equatio (3-4) give m= D D (3-6) ( )( ) A B A B. Here we must otice that µ = is tetatively adopted i the diffusio uctio depth u = µ D γ τ i the preset case. Therefore the µ value i the relatio of m= ( D A D B)( A B) µ must be cocretely ivestigated so as to be suitable for the cocered iterdiffusio problems sice the m value is experimetally obtaied Uified Theory of Diffusio Problems The experimetal results show that the relatio of v RP = is valid i the metallurgy field. This meas that the shape of diffusio system is uchageable before ad after diffusio treatmet. that case Equatio (-9) is thus rewritte as τ = ( D ). (3-7) Equatio (3-7) of ( τ ) is thus equivalet to Equatio (-) of ( ) tx. However the diffusio flux of Equa- 7

10 tio (-) is rewritte as ( τ ) = ( τ ) + ( τ) + ( ) J J J J R F R eq JF τ = D. the followig various diffusio problems are systematically ivestigated by applyig the diffusio flux of Equatio (-8) to them for = uder the coditio of Equatio (-8) is the expressed as ( )( ) v D D t. QP = A B A B ( i ) JQ ( t x ) JF ( t x ) vqp ( t x ) Jeq JF ( t x ) D x ( t x ) ( ) J ( t x ) = J ( t x ) + v ( t x ) + J J ( t x ) = D ( t x ) = + + = ii Q F QP eq F. (a) terdiffusio: Uder the coditio of D = D = D Equatio (3-8) becomes J t x = D t x + v t x + J (i) ( ) x ( ) ( ) (ii) ( ) ( ) ( ) Q QP eq J t x = D t x + v t x + J. Q x QP eq (b) Oe-way diffusio: Uder the coditio of D = Equatio (3-8) becomes (i) ( ) ( x ) ( ) (ii) J ( t x ) = v ( t x ) + J. J t x = D t x + v t x + J Q QP eq Q QP eq (c) mpurity diffusio: Uder the coditio of D = ad (i) J ( t x) = D ( t x ) + J (ii) ( ) Q (d) Self-diffusio: Uder the coditio of x eq A B A = Equatio (3-8) becomes Q = = ad (i) JQ ( t x ) = Jeq (ii) JQ ( t x) = Jeq f we rewrite J ( t x ) ad J ( t x) Q Q. Q as B J t x = J. eq = = Equatio (3-8) becomes A B ( k ) Q( ) ad JQ ( t x) JQ( t x) J t x J t x for = k = k (3-8) the above theory is still valid i the elemets system. the diffusio problems it was thus foud that the behavior of diffusio regio space plays a importat role for uderstadig diffusio mechaisms Self-Diffusio Theory The diffusivity expressio of Equatio (-5) is valid regardless of pure materials [3]. The Eistei theory of Brow motio shows that the self-diffusio of water is visualized by diffusio pheomea of polle micro particles. t is thus obvious that the self-diffusio occurs i a pure material. However eve if we apply the F law to the self-diffusio problems i a pure material we caot uderstad whether the self-diffusio occurs or ot sice the cocetratio is uchageable i the self-diffusio system. Lagevi theoretically ivestigated ot the diffusio equatio of collective motio but the motio equatio of a diffusio particle with a viscosity resistace i a pure liquid material [7]. t was thus revealed that the diffusio particle moves i accordace with the ormal distributio rule. The theories of Lagevi as well as Eistei are relevat to diffusio particles with a drivig force i a pure material. other words the diffusio pheomea of a pure material itself are ot directly ivestigated. However Equatio (-5) shows that the self-diffusio occurs i a pure material eve if there is o drivig force i the diffusio system. The self-diffusio problems have bee also experimetally ivestigated as a impurity diffusio problem by 8

11 usig small quatities of the isotope material. Strictly speakig however the self-diffusio problems are obviously differet from the impurity oes as ca be see from the differece of (i) betwee (c) ad (d) i Sectio 3.3. As discussed i Sectio. the correspodig F law to the F law is icomplete without the itrisic diffusio flux. the followig therefore the self-diffusio problems of a pure material are directly ivestigated i accordace with the itrisic diffusio flux regardless of a drivig force. () Applicatio of itrisic diffusio flux to self-diffusio problem We caot experimetally cofirm J eq i Equatio (-3). The theory of Brow motio reveals that the diffusio pheomea occur i a material eve if the macro cocetratio gradiet is zero. The cocept of itrisic τ is for covei- diffusio flux should be thus valid i a localized regio space. The coordiate otatio ( ) ece used istead of ( t x ) ad/or ( tx ) i the followig. Usig the coordiate otatio ( τ ) for the equatio of J ( t x ) = J relatio of { ( ) ( )} + eq = + Q eq relevat to the self-diffusio the J τ τ τ τ d (3-9) is obtaied. f the cocetratio differece betwee ( τ ) ad ( τ τ ) Equatio (3-9) is rewritte as Takig accout of ( τ τ ) Jeq τ + + is defied as =. (3-) + i Equatio (3-9) ad usig Equatio (-5) for Equatio (3-) we have the relatio of = J. eq D (3-) Equatio (3-) shows a istataeous cocetratio gradiet i the localized time τ ad space i the cause of the thermal fluctuatio eve if the macro cocetratio gradiet is zero. other words although the cocetratio gradiet of a pure material is zero i macro behavior it is ozero i the miute regio. The behavior of cocetratio distributio i the local space is thus obtaied as J = +. (3-) D eq ( ) ( ) a pure material or a material i the thermal equilibrium state Browia particles move i accordace with Equatio (3-). f J eq = the Browia particles get a stadstill state icosistet with the Eistei theory of Brow motio. The itrisic diffusio flux J eq is thus a importat cocept i the fudametal diffusio theory. At the same time the F law is icomplete without J eq. f we apply Equatio (3-) to the iterdiffusio problems the relatio of ( ) = Jeq ( ) D + (3-3) = = = is valid i accordace with Equatio (3-3). Equatio (3-3) reveals that Equatio (3-6) is valid because of ( ) = ( ) =. = = Usig the otatio α = Jeq D for Equatio (3-) the cocetratio behavior of a elemet i the thermal equilibrium state is expressed as uder the coditio of = ( ) α ( ) = + (3-4) α = i the localized space. The diffusio regio of self-diffusio problem of a pure material is for coveiece divided ito regios as a biary system iterdiffusio problem composed of a pure material of < ad the pure material of 9

12 > at τ =. The diffusio regio is betwee A B at a time τ the. The boudary values are A = for A ad B = for B uder the coditio of =.5. Usig Equatio (3-4) for = the relatios of are obtaied where ( ) ( ) eq ( ) = α + ad ( ) α = J D. Equatio (3-5) shows that ( ) is valid if. f we pay attetio oly to oe side of = α + (3-5) ad ( ) or deped o ad that it is obvious that the particle migratio occurs via the radom movemet i the pure material. other words the diffusio occurs i a pure material eve if the macro cocetratio gradiet is zero. O the other had if we pay attetio to both sides of ad i Equatio (3-5) ( τ ) ( τ ) = = seems to be valid sice we caot actually distiguish their diffusio particles betwee A < < B whether they were oes of or i the iitial state at τ =. () Physical meaig of itrisic diffusio flux Here we estimate the J eq value i the followig. Whe the averaged ump distace or the averaged collisio distace is writte as l i a pure material the substitutio of A = l ad B = l ito Equatio (3-5) yields Jeq = Dself l (3-6) where D is rewritte as D self ad l correspods to the cocetratio gradiet. t is cosidered that the order of l is several agstroms ad the diffusio uctio depth i a usual experimet is several micrometers. Therefore the absolute value of JF = D is cosiderably small compared with that of Jeq = Dself l. We caot actually observe J eq i the macro state because of Equatio (3-6). However we ca approximately estimate it by usig the relatio of J D l * eq = self * where D self is a self-diffusio coefficiet obtaied from the tracer diffusio experimet for the cocered material =.5 ad l correspods to the averaged ump distace or the averaged collisio distace of the diffusio particle. Equatio (3-5) idepedet of τ meas the time-averaged cocetratio profile caused by the Brow motio i the localized regio space. The diffusio occurs eve i a pure material resultig from the chai reactio of istataeous umpig of diffusio particle ito micro holes caused by a thermal fluctuatio i a material miute regio. Equatio (3-6) thus gives evidece that the Brow motio occurs i a pure material eve if a exteral force does ot exist i the diffusio system. From the historical poit of view if the F law were ivestigated at a early stage i accordace with the Gauss divergece theory we might uderstad the Brow motio behavior before the Eistei theory [] [3] [4] Diffusio Equatio of Micro Holes The space coordiate depedece of micro holes has bee eglected i the diffusio theory metioed above. the metallurgy field however the experimetal results of multiple markers show that their movemets deped o the space coordiate i accordace with a marker positio set i the iitial diffusio couple [8]-[3]. This idicates that we must ivestigate the space coordiate depedece of micro holes i the diffusio study i detail sice the behavior of micro holes is visualized by the characteristic of iert markers. that case Equatio (3-) is rewritte as where H is the ormalized cocetratio of micro holes betwee xa x xb. geeral there is a differ- H H H ece ( A B ) H + = (3-7) = H = betwee the thermal equilibrium cocetratio of micro holes ( x x ) A A i the

13 H material A ad that of ( x x ) T. Okio H B B i the material B. Whe the differece is ot egligible the diffusio of micro holes occurs betwee xa x xb. H For the thermal equilibrium cocetratio of micro holes i the diffusio regio xa x xb the geeratio/aihilatio term of the diffusio equatio is expressed as H H ( ) k (3-8) H where k H is a chemical reactio costat. Takig accout of Equatio (3-8) the diffusio equatio of micro holes is expressed as ( ) ( ) = + (3-9) H H H H H t x D x kh H where D is a diffusivity of micro holes betwee xa x xb. The K-effect shows that the cocetratio of micro holes yields a supersaturated sectio ad a usaturated oe i the diffusio regio. Therefore the averaged value of diffusio flux yieldig xb H H ( ) ( ) x x k x (3-3) B A H d xa is cosidered to be egligible. We thus assume that the averaged cotributio of the geeratio/aihilatio term to the total diffusio flux i the diffusio system is egligible. Here if we itegrate Equatio (3-9) with respect to x the diffusio flux relatio of J H = H dx= D H H k H H dx D H H ( ) (3-3) x t x H x is obtaied by eglectig the cotributio of Equatio (3-3). that case Equatios (3-7) (-) ad (3-3) yield + H x D x x ( D D ) x = = = = uder the coditio of + = H. Equatio (3-3) is thus rewritte as H D = D = D = D = = D. (3-3) Equatio (3-3) shows that we ca accept a micro hole as a virtual diffusio particle. The diffusio equatio of H i the elemets system iterdiffusio becomes t = x( D x ) (3-33) usig a commo iterdiffusio coefficiet D. t seems that Equatio (3-33) is a idepedet equatio of. H However it is ot idepedet because of Equatio (3-7). Whe we must ivestigate the ifluece of o the iterdiffusio problems i detail the diffusio equatio of micro holes yieldig H H H H = D + k (3-34) t x x ( ) H( ) should be ivestigated. Equatio (3-34) shows that the behavior of multiple markers depeds o the time ad H space sice the obtaied depedet o the time ad space is visualized by the iert markers Applicatio of Preset Theory to Actual Diffusio Problems Hereibefore we discussed some fudametal problems i the bulk diffusio of a ideal diffusio couple. The iterdiffusio problems for a may elemets system such as a alloy a compoud semicoductor a multilayer thi film a microstructure material ad so o have bee widely ivestigated [4] [3]-[34]. t is the fudametally idispesable for the material sciece ivestigatio to uderstad the ( ) elemets system iterdiffusio theory sice the migratio of each elemet i a material occurs durig the thermal treatmet i the material fabricatio process. the diffusio study history the cocept of itrisic diffusio has bee accepted i order to uderstad the K effect. the existig theory of iterdiffusio problems the Darke equatio derived from the itrisic diffusio cocept has bee widely used for aalyzig them [35]. The applicatio of the Darke theory to a may elemets system iterdiffusio was extremely complicated eve if it was a terary elemets system [36]. O the other

14 had the simple relatio amog iterdiffusio coefficiets obtaied here will be extremely useful for aalyzig various iterdiffusio problems. As ca be see from the Appedix [B] we revealed that the Darke equatio is ot mathematically valid. t is also reported that the aalytical results obtaied by usig the Darke equatio are ot cosistet with the experimetal results [37]. This fact is cosistet with the preset theory where there is o such cocept of itrisic diffusio i the diffusio pheomea. Uder the coditio of Equatio (3-7) therefore it is ecessary for aalyzig a may elemets system iterdiffusio problem to solve the diffusio equatios (3-33) ad (3-34) of each elemet i accordace with their iitial ad boudary coditios. Here it seems that these diffusio equatios are idepedet of each other sice they have a commo iterdiffusio coefficiet. However it is ot idepedet because of Equatio (3-7). the parabolic space discussed later the aalytical method of a oliear diffusio equatio is established. The aalytical solutios of Equatios (4-48) ad (4-49) are thus obtaied the. Usig them for aalyzig each of the above equatios we ca fudametally solve a ( ) elemets system iterdiffusio problems. The theory discussed here is thus widely applicable to aalyzig the fudametal problems of elemets system iterdiffusio. Sice the preset aalytical theory is fudametal it will be ecessary for actual diffusio problems to modify it partly i accordace with the cocered problems like the Boyle harles law for a ideal gas state is modified ito the va der Waals equatio i the actual case by icorporatig a iteractio betwee gas particles ito it. 4. Aalysis of Diffusio Equatio i Parabolic Space Eistei theoretically revealed that the Browia particles radomly move i accordace with the parabolic law [3]. After that it was experimetally cofirmed by Perri [38]. The parabolic law is uiversally show i pheomea relevat to the ormal distributio i a probability problem. The Brow problem is a study subect ot oly i the material sciece but also i the complex-system scieces relevat to the Markov process. Therefore the aalytical method of the Brow problem is mathematically commo i various sciece fields. The diffusio uctio depth is directly relevat to the parabolic law. The diffusio problem is ustifiably oe of the Brow problems. This idicates that the solutios of diffusio problems are possible i the parabolic space. other words the aalytical method discussed here is widely applicable to various Brow problems. The F law is a cotiuous equatio i the coservatio system uder the coditio of o sik ad source of diffusio particles ad it is oe of the basic equatios i physics. Whe the existece of a drivig force or the sik ad/or source of diffusio particles is egligible i a give diffusio system Equatio (-6) is a parabolic type liear partial differetial equatio i the evolutio equatios. A lot of diffusio problems have bee ivestigated by solvig the diffusio equatio of the time ad space ( txyz ) sice 855. Whe the diffusivity depeds o the cocetratio the diffusivity becomes a fuctio of the idepedet variables ( txyz ) via the cocetratio. Eve if the diffusio equatio depeds oly o ( ) tx the mathematical solutios of the oliear partial differetial equatio are almost impossible. A ew aalytical method i the parabolic space which is extremely superior i aalyzig to the existig oes is discussed i the followig. 4.. Defiitio of Parabolic Space = t ad τ = t uder the codi-. The relatios of differetial operators are obtaied as For coveiece hereafter we use the coordiate otatio θ( ) ( 3) space ( txyz ) ito the parabolic space ad they are correlated to x tio of ( tx x x) ( txyz ) ad 3 = trasformed from the time ad τ x x x x x 3 = = τ 3 τ τ 3 = = t t τ t t t 3 τ τ 3 From the similar calculatio the differetial operators i the parabolic space are obtaied as

15 where = ( 3) = σ τ τ = = ad = { } σ σ = θ t τ τ because of the Hermite characteristic. σ (4-) 4.. Diffusio Equatio i Parabolic Space f we directly apply Equatio (4-) to homogeeous partial differetial Equatio (-6) the ellipse type differetial equatio is obtaied as where τ ( 3) f = for θ( ) θ σ( 3) = σ D σ( 3) (4-) = is physically valid i accordace with iitial coditios []. Equatio (4-) becomes ( ) d( ) d d D = d d d which is the well-kow Boltzma trasformatio equatio []. Further it is rewritte as dd d ( ) d ( ) = + D D d d d The itegral calculatio of Equatio (4-4) yields the itegro-differetial equatio give by where J D( ) ( ) =. = Here if we defie the expressio of D ( ) d ( ) = Jexp d η d D( η) J ( ). (4-3) (4-4) η (4-5) η D( η ) (4-6) = Jexp d η J ( ) D( ) d ( ) = (4-7) d is obtaied from usig Equatio (4-6) for Equatio (4-5). Boltzma trasformatio equatio (4-3) is essetially equivalet to Equatio (4-7). We ca accept Equatio (4-7) as a diffusio flux i the parabolic space. The F law i the time ad space caot be used directly for aalyzig a diffusio problem. However the diffusio flux of Equatio (4-7) is possible for aalyzig a diffusio problem uder the iitial coditio of Equatio (4-6). Further as described later the solutios of three dimesios problems i the parabolic space ( 3) are obtaied as a liear combiatio amog the solutios of each oe dimesio diffusio problem. Therefore Equatio (4-7) is the most importat essetial equatio relevat to the diffusio problems i the parabolic space. the diffusio flux is defied as the parabolic space ( ) 3 where the compoet of J( 3) ( J J J ) ( ) ( ) J = D (4-8) 3 σ 3 = is expressed as 3 J η = J exp d η D for η = (4-9) 3

16 usig J D ( ) = for = 3. Here ote that the iitial ad/or boudary coditio is take ito = accout i the diffusio flux of Equatio (4-6) or (4-8) although it is ot take ito accout i the F law. is equivalet to Equatio (-6) i the time ad space Equatio (4-8) i the parabolic space ( 3) ( txyz ). mathematics the followig relatio of d = ( ) + D for = 3 d D (4-) is geerally valid if the diffusivity D depeds o idepedet variables. Equatio (4-) is extremely useful for aalyzig a oliear diffusio equatio Aalytical Solutios of Liear Diffusio Equatio The utility of the preset method is cofirmed through solvig some typical diffusio problems by cocrete calculatios. The mathematical method used here is also widely applicable to the Brow problems i various sciece fields. () Oe dimesio parabolic space We first solve the liear diffusio equatio give by uder the iitial ad boudary coditios of The iitial coditio of Equatio (4-) is rewritte as t ( ) ( ) tx = D tx (4-) ( ) A ( ) ( ) ( ) t = x< : x = x> : x = B t > t = A t = B. ( ) = A ( ) B x (4-) = (4-3) i the parabolic space. Substitutig D= D ad = ito Equatio (4-5) the iitial value of ( ) ( ) d ( ) J = D for = (4-4) d = is obtaied. Usig Equatio (4-4) for Equatio (4-5) the ordiary differetial equatio of ( ) ( ) d d 4 = exp D is obtaied. From the defiitio of the well-kow error fuctio the geeral solutio of Equatio (4-5) is obtaied as (4-5) ( ) ( ) = πd erf + γ (4-6) D where γ is a itegral costat. Usig the iitial coditio of Equatio (4-3) for the geeral solutio the well-kow solutio is obtaied as A B A B ( ) + = erf. D (4-7) the aalytical method i the parabolic space the solutio is thus easily obtaied oly by the elemetary itegral. t is therefore obvious that the preset aalytical method is extremely superior i aalyzig diffusio equatios to the existig methods. () Parabolic space of dimesios 4

17 case of D= D Equatio (4-9) is rewritte as ( ) J = D exp 4D Usig this equatio for = 3 the relatio of for ( ) ( ) = ( ) exp 4D ( ) σ ( ) ( ) = exp 4D ( ) 3 exp 3 4D is obtaied from Equatio (4-8). The operatio of σ o both sides of Equatio (4-8) yields the Poisso equatio give by D =. (4-8) ( ) ( ) = exp 4 D. (4-9) = = We solve the ellipse type differetial equatio (4-9) correspodig to Equatio (-6) uder the iitial ad boudary coditios i the followig. The iitial ad boudary values of Equatio (-6) give by correspod to i the parabolic space ( ) ( ) ( ) ( ) ( ) 3 txyz = for t y> z> ad x= txyz = for t x> z> ad y= txyz = for t x> y> ad z= txyz = for t= x> y> ad z> ( ) ( ) ( ) ( ) 3 (4-) = = = = (4-). 3 the aalysis of Equatio (4-9) it is easily foud that the equatio of ( ) S ( ) = π erf = D D (4-) satisfies Equatio (4-9). other words Equatio (4-) is the particular solutio of Equatio (4-9). Whe the geeral solutio of Laplace equatio of is obtaied as ( ) ( ) L i ( ) = (4-3) i= = the geeral solutio of Equatio (4-9) is expressed as i accordace with the mathematical theory. Assumig the equatio of ( ) ( ) ( ) = + (4-4) L S ( ) F ( ) = (4-5) L = ad usig the variable separatio method the solutio of Equatio (4-3) is possible. Substitutig Equatio (4-5) ito Equatio (4-3) the relatio of = F ( ) d F d ( ) = 5

18 is valid. order to satisfy this equatio for a arbitrary must be valid uder the coditio of d d the equatio of λ F( ) = (4-6) λ =. (4-7) = Equatio (4-7) yields λ = for = ad solutio of Equatio (4-6) is expressed as L ( ) = A + A + ( ) where A ad A are arbitrary costats. ( ) + A + = ad A = = πd are determied by the iitial coditio. The well-kow solutio of Equatio (4-9) is thus obtaied as case of = 3 ad = ( ) = ( erf ) D. (4-8) λ substitutig the solutio of Equatio (4-6) give by λ ( ) = + e + e λ A A ito Equatio (4-5) the geeral solutio of Equatio (4-3) is obtaied as L A e A e λ = λ λ ( ) = ( + + ) (4-9) where A+ ad A are arbitrary costats. However there is o solutio satisfyig the iitial ad boudary coditios because of Equatio (4-7). As a result the geeral solutio of Equatio (4-3) is obtaied as A A ( ) = ( + + ) (4-3) L λ = uder the coditio of λ = λ = λ 3 =. Sice the iitial ad boudary coditios yield A + = the preset solutio of Equatio (4-3) is thus obtaied as where ( ) ( ) = L = = πd i accordace with the iitial ad boudary coditios. The solutio of Equatio (4-9) is thus obtaied as f we use erfc( η) erf ( η) ( ) = ( ) = erf D. (4-3) = for = 3 the solutio is also preseted as x y z ( t x y z) = erfc + erfc + 3 erfc. Dt Dt Dt (4-3) The preset study reveals that the particular solutio is easily possible ad the complemetary fuctio of Equatio (4-9) is a costat value to satisfy the give iitial ad boudary coditios because of the sigular characteristic of iitial coditios. other words the diffusio behavior is icorporated ito the ihomogeeous term of the Poisso equatio. As a result the solutio is obtaied as a liear combiatio of the error fuctios. Although the diffusio problems have ot ever bee ivestigated i the parabolic space it is obvious that the preset solutio is thus exceedigly simple ad elegat compared with the usual solutio of 6

19 for µ 3 = k where 3 = 3 µ t kx kx ( ) = e ( + e + e ) txyz A A µ = x = x x = y x = z. (4-33) (3) Aalysis of ihomogeeous diffusio equatio The research subects of liear diffusio pheomea i detail are ofte expressed by a ihomogeeous partial differetial equatio. Therefore the applicatio of the preset method to those problems is ivestigated. Geerally such liear diffusio equatios are expressed as where L is the liear operator of ( ) Ly = g t x y z (4-34) L = D (4-35) t. From the mathematical theory the solutio of ihomogeeous Equatio (4-34) is obtaied as where f ( ) by ad f( ) ( ) ( ) y= f txyz + f txyz (4-36) S txyz is the complemetary fuctio i.e. the geeral solutio of homogeeous equatio give S txyz is a particular solutio of Ly = (4-37) ( ) ( ) fs txyz = L gtxyz. (4-38) Equatio (4-) shows that the differetial operators i the parabolic space are ot able to apply to Equatio (4-34). However the liear operator i the parabolic space give by LP = θ σ + D σ σ (4-39) is possible for aalyzig the homogeeous Equatio (4-37). other words we ca obtai the complemetary fuctio by the preset method. order to obtai a solutio of a ihomogeeous diffusio equatio therefore we must obtai a particular solutio by the existig methods Aalytical Solutios of oliear Diffusio Equatio Usig the experimetal profile ( ) for Equatio (4-3) Matao obtaied the D ( ) profile i the iterdiffusio problems betwee solid metals i 933 [6]. The empirical Boltzma Matao method has bee widely applied to the aalysis of the iterdiffusio experimets betwee solid metals. However the mathematical solutios of Equatio (4-3) had ot yet bee obtaied sice 894 for such a log time util the recet work [9]. f the aalytical solutios of Equatio (4-) or (4-8) obtaied we ca uderstad the effect of a drivig force o the diffusivity. other words D df of Equatio (-7) will be aalytically obtaied. () Aalytical method i parabolic space Whe the diffusivity is affected by a drivig force the mathematical solutios of the oliear equatios (-) ad (4-3) or (4-7) are impossible if o other relatio betwee diffusivity ad cocetratio is give. that case the diffusivity depeds o the idepedet variables ad Equatio (4-) is geerally valid i mathematics the. Equatios (4-7) ad (4-) are thus simultaeously solved i the followig. The itegro-differetial equatio (4-5) is superior i the approximate calculatio to the secod order differetial equatio (4-3). For example usig a effective diffusivity D eff for the expoetial part i Equatio (4-5) the relatio of η dη = D 4D ( η ) is valid because of the characteristic of the itegral calculatio where D is rewritte as a iterdiffusio coeffi- eff 7