Elements of Kinetic Theory

Transcription

1 Eements of Kinetic Theory Diffusion Mean free path rownian motion Diffusion against a density gradient Drift in a fied Einstein equation aance between diffusion and drift Einstein reation Constancy of chemica potentia uiding of potentia barriers Other transport phenomena Heat transport Momentum transport=viscosity Mosty in Kitte and Kroemer Chap. 14 Phys 11 (S009) 8 Kinetic theory 1

2 How do systems come in equiibrium? Energy transfer If gases at different temperatures are put in contact, moecues of the hotter gas have in average higher energy and transfer net energy to the ower temperature gas => temperature equiibrium. Energy transport by diffusion. Not instantaneous! => therma conductivity Heat transfer equation Simiary, transfer between wa of gas encosure and gas. Momentum transfer Phys 11 (S009) 8 Kinetic theory =>e.g., back body radiation: equiibrium between was and photons inside cavity if shear between fuid voumes Partice transfer reated to viscosity (see ater) If a gas system 1 is put in contact with another gas system where the concentration of gas moecues is ower, the higher density in system 1 wi favor diffusion of moecues to system => concentration equiibrium => Diffusion equation

3 Scattering Mean Free Path Interaction cross section σ Consider a beam of partices incident on a target Probabiity of interaction in a sab of thickness dz = σ n dz dz Mean free path Exampe: hard spheres Probabiity of interaction in interva dz Surviva probabiity = 1 σn Partice enters medium at z=0. is the attenuation ength N( z dz) = N ( z) 1 dz P( z dz) = P( z) 1 dz Probabiity of interaction between z, zdz d dz d Cross section : dimension = area σ = πd The surviva probabiity varies as: ( ) dp dz = P z Prob ( z )dz = exp z dz P z ( ) = exp z Phys 11 (S009) 8 Kinetic theory 3

4 Diffusion: No Concentration Gradient rownian motion Succession of scatters:consider a partice of speed v Assume isotropic scattering, no concentration gradient => Average dispacement between two scatters aong z axis θ δz between scatters = scosθ e s ds d cosθ dϕ π = 0 Phys 11 (S009) 8 Kinetic theory z constant concentration => does not depend on θ Average dispacement squared between two scatters aong z axis = Variance δz between scatters = ( scosθ) e s ds = Diffusion coefficient 4 d cosθ dϕ π = 3 Number of scatters for partices of speed v per unit time => Evoution of variance with time d δz fixed v = v dt 3 Average on distribution of veocities d δz = v dt 3 = vf ( v)dv 3 f ( v)dv D = v 3 = D = d δx dt = d δy dt dn scatters dt = v

5 Expectation vaues δz = δz int interactions δz int δz = N int σ δz = σ δz d δz dt dσ δz dt interactions = d N int dt = v σ δz int Genera Method int δz int = N int σ δz Independence of scatters int = v δz int How to compute expectation vaues - Fix veocity - Average on ength s before interaction surviva probabiity ds - Average on ange (assumption of isotropic distribution) - Mutipy by number of interactions per unit time v - Average on veocity s m e 0 1 cos cos s o ds m = m! o o θ d cosθ = 0 θ d cosθ = 1 3 Phys 11 (S009) 8 Kinetic theory 5

6 Diffusion: Concentration Gradient Suppose that we have concentration gradient aong the z axis s to first order in s dn n 0 dz 1 = σn 1 = dn o o n o dz z = dn o o n o dz scosθ => depends on θ θ z => Probabiity of surviva aong direction θ is such that P surviva ( s ds) = P surviva ( s) 1 ds dp surviva ( s) = P surviva ( s) 1 1 dn o o n o dz scosθ ds P surviva ( s,cosθ) = exp 1 s s 1 dn o o n o dz cosθ e => Probabiity of interaction between s and sds s o 1 s 1 dn o o n o dz cosθ P interact = P surviva ( s) ds = ds s e o 1 s 1 dn o o o n o dz cosθ 1 1 dn o n o dz scosθ The mean dispacement aong the z axis between coisions (keeping ony first order in reative gradient) ds s δz between coisions = scosθ e o 1 s s 1 dn o o o n o dz cosθ d cosθ dϕ π Phys 11 (S009) 8 Kinetic theory 6

7 Diffusion: Concentration Gradient taking into account that we get s m 0 s e o ds m = m! o o δz between coisions = 1 3 o ( 3 o ) 1 dn o n o dz = o 3 1 dn o n o dz Each partice undergoes v/ scatters per unit time. Hence, the mean transport veocity aong the concentration gradient is dz dt v = δz between coisions = o v o 3 Averaging on veocity distribution d δz = w dt z = v o 1 dn o 3 dz = D 1 dn o n o dz n o 1 dn o n o dz Phys 11 (S009) 8 Kinetic theory 7

8 Transport Diffusive transport => Partice fux J z = n o w z = D dn or more generay dz Fisk s aw: Opposite to gradient This is one exampe of transport: In addition to random veocity v there is a coherent transport (or drift) veocity w and net fuxes of partices Simiar transport of charged partices expains eectric mobiity, of energy expains heat conduction, of momentum expains viscosity. Conservation of the number of partices n Consider a voume V. The decrease of the number of partices inside voume has to be equa to the tota partice fux through the surface. n d 3 x = t J ds V S but by the Divergence Theorem J d S = J d 3 x This has to be fufied whatever the voume. S V d S Hence the partice conservation equation: Diffusion equation Repacing J by its vaue J = D n n t J = 0 J = D n we get Phys 11 (S009) 8 Kinetic theory n t = D n 8 wave equation 1 c A t = A

9 Drift in an Eectric Fied Charged Partices (eectrons, hoes, ions) Drift Veocity Consider an eectric fied aong the z axis. In addition to its random veocity, each partice wi acquire a net veocity in z direction from acceeration between dz dt E coisions It is advantageous therefore to work with the time δt before the next coision instead of s. If the coision time τ c =/v is constant, the probabiity of coision between δt and δt dδt is Mean dispacement between coisions: δz co. = δt τ e c Each partice undergoes v/ coisions per unit time => => Averaging on random veocities δv z = qe m δt δz = v cosθδt 1 qe m δt dδt τ c = δz between coisions v = qe m e δt / τ c dδt τ c surviva probabiity d cosθ v dϕ π v cosθδt 1 qe m δt = qe m τ c = qe mv w z = qe m v = qe m τ c =acceeration x τ c Note: this stricty appies to case where coision time τ c =/v is constant. Otherwise w = qe z m 3 v 1 = qe 3 v m τ c eff Phys 11 (S009) 8 Kinetic theory 9

10 Mobiity Drift in an Eectric Fied () w = µ E Constant τ c =/v Ohm s aw µ = q m τ c Consider a wire of ength L and sectiona area A. If the wire is thin enough L A E = V where V is the appied votage L I = naqw = na q m τ V c L V = L 1 A nq m τ c I conductivity σ c Phys 11 (S009) 8 Kinetic theory 10

11 Constant coision time Consider the ratio Genera case for constant τ c =/v Einstein Reation qd µ = m τ c v 3 = m τ v c τ c 3 for constant τ c = /v D = ττ c m = k Tµ q = 1 / mv 3 If the fied is ow enough for the partice to remain in therma equiibrium at temperature τ = k T, it can be easiy shown by integrating by part the integra giving <v> that D = τ m 3 v 1 3 d dv = ττ c eff m = τ µ q = τ = k T We then sti have D = ττ c eff m = k T µ q Phys 11 (S009) 8 Kinetic theory 11

12 Constancy of the tota chemica potentia aance between Eectric Potentia and Density Gradient Consider a charged partice in an eectric fied aong Oz=> For constant τ c =/v, this induces a drift veocity which wi increase the concentration aong w z Any density gradient wi induce a diffusion such that Inversey a gradient of charged partices wi induce an eectric fied which wi create a drift veocity These two contributions wi baance when these two veocities are opposite => at equiibrium integrating to have the potentia we get w z = D 1 n w z = µ E dn dz = τ µ q µ E = τ µ q qe τ 1 n n 1 Remembering that the interna chemica potentia is we concude that the tota chemica potentia is constant 1 n 1 n dn dz dn dz dn dz = 0 V(z) = qv(z) τ og n( z) = constant z Edz z o µ int z ( ) = τ og n( z) n Q qv (z) µ int ( z) = constant Phys 11 (S009) 8 Kinetic theory 1

13 aance between drift and diffusion Exampes: attery (K&K p.19) Consider an eectroyte A: ions A - negative ions N p-n Diode A A A A A A A P In the midde of the ce, equiibrium between positive and ut on eectrodes, difference of behavior => seective depetion repusion. E = ρ εε o Stops A - neutraization E A N AN e P e P If the two eectrodes are not connected N P x V = E.dr V N P ε F ( p) p hhhhhhhhhh eeeeeeeeeeeeee n ε F ( n) ε F p eeeeeeeeeeeeee ( p) hhhhhhhhhh n ε F ( n) On n side, the donors give their eectrons and positive charges remain behind On p side, the acceptors capture the eectrons, generating fixed negative charges. The resuting fied generate a potentia barrier which prevents current to fow in one direction Phys 11 (S009) 8 Kinetic theory 13

14 Cacuation Method Consider species of opposite charge q ± : number density n ± ( x) Combine µ int ( x) q n ( x)φ( x) = Constant µ int and. E = Φ = q n x ( ) q n ( x) ( x) q n ( x)φ( x) = Constant κε o 3 equations for 3 functions n n Φ (κ,often aso written ε, is the reative dieectric constant of the medium) Note that in semiconductor books, the constancy of the chemica potentia is expressed in terms of the sum of the drift and diffusion currents being zero. This is the same physics expressed in different ways! Phys 11 (S009) 8 Kinetic theory 14

15 Concentration Gradient & Gravitationa Fied Simiary In the earth gravitationa fied the down drift from the potentia w z = gτ c The diffusion upwards is w z = D 1 dn dn n dz dz > 0 The two veocities shoud be equa in magnitude and opposite D 1 dn n dz = ( gτ c ) Using the Einstein equation D = ττ c m τ 1 dn n dz gm = 0 or τ og n ( z ) gmz = constant n 1 The sum of interna chemica potentia and externa potentia per partice is constant! g z Phys 11 (S009) 8 Kinetic theory 15

16 Energy and Momentum Transfer Coision between partices of equa mass m v 1 The center of mass veocity is and veocities in center of mass. Phys 11 (S009) 8 Kinetic theory Consider a partice of veocity v 1 coiding with another one of veocity v (non reativistic) v cm v * v 1 v * Lab frame v cm = v * 1 = v 1 v cm = v 1 v v 1 v As energy is conserved, after the scatter the partice veocities are v 1 v v '* 1 = v '* = where u is a random vector of unit ength=> in aboratory frame ' v 1 = v '* 1 v v cm = 1 v v u 1 v v 1 ' = v 1 v 4 u v v v 1 * Center of mass frame v * v = 1 v u v 1 v u v * v 1 v

17 Energy and Momentum Transfer This has to be averaged on random directions of u. u = 0 ' v v 1 = 1 v = ' v v ' v 1 = 1 v 4 => Averaging out of momentum and energy! Energy transfer 1-> ΔE = 1 m v ' v = 1 m v 1 v u v 1 v Δ p = m Consequence : heat conduction viscosity 4 = v 1 v = v ' Momentum transfer v ' v v = m 1 v Phys 11 (S009) 8 Kinetic theory 17

18 Therma Conductivity Consider a medium in oca therma equiibrium but with a therma gradient aong z. Diffusion wi transport energy from hotter region to cooer regions: Consider a partice 1 which just has been scattered: its initia veocity is v 1 and ange θ,ϕ. At the next coision with partice after path s, it wi transfer in average if partice 1 comes from region of temperature T 1 and partice comes from a region of temperature T. The mean energy transport aong z per coision is 3 < Average energy transfer δz >= k T 1 T ( ) s scosθ e ds with T 1 T = T T δz = z z scosθ 3 < Average energy transfer δz >= - k T ( scosθ) s e ds dv cosθ z Taking into account the tota number of coisions per unit time we obtain the energy fux aong z (averaged over v) is J Qz = 3 nk T v or J Q = κ T z 3 therma conductance κ = 3 nk v 3 = C v 3 = CD Phys 11 (S009) 8 Kinetic theory 1 m v 1 v = 3 k T 1 T 18 d cosθ dϕ π dϕ π = - 3 k T z 3 where C is the heat capacity per unit voume

19 Therma Conductivity () Heat equation CδT = δu where u is the energy density The oca increase of temperature with time is y same argument of energy conservation => T t = 1 C κ T = D T This is the diffusion equation again! T t = 1 u C t u t J Q = 0 Phys 11 (S009) 8 Kinetic theory 19

20 Phys 11 (S009) 8 Kinetic theory Veocity gradient :Viscosity Consider a medium with a buk fow w aong x added to the random therma veocity v r w x. Assume a buk veocity gradient aong z. Consider a partice which just has been scattered: its initia veocity is v and ange θ,ϕ. At the next coision after path s, it wi transfer in average a momentum aong the x axis: Δp x = m v 1x v x = m w 1x w x The mean momentum transported aong z between coisions is: s Δp x δz = m Taking into account the number of coisions per unit time, the momentum fux is J z,p x = mn v w x 3 z = η w x z Notes: More generay Shear force ( ) = m w x with m w 1x w x w 1x w x scosθ e J i, j = η w j x i 0 ds z scosθ (tensor!) d cosθ dϕ π ΔP x δz = m w x z 3 and averaging over v exampe: aminar fow in a tube. The fuid is at rest at the wa. The viscosity ead to pressure drop! Independent of pressure ρ p D 1/ p (arger mean free path=>arger momentum difference) In addition buk viscosity reated to expansion 'dynamic' pressure P=ζ w (ζ usuay much smaer than η) v with shear viscosity η = mn v 3 = ρ v 3 = ρd J p = η ( w )