Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 4 potx

4.1 Random Walk and Diffusion 61  R 2  = n  i=1  r 2 i  +2 n−1  i=1 n  j=i+1  r i r j  ,  X 2  = n  i=1  x 2 i  +2 n−1  i=1 n  j=i+1  x i x j  . (4.20) The first term contains squares of the individual jump lengths only. The double sum contains averages between jump i and all subsequent jumps j. Uncorrelated random walk: Let us consider for the moment a random walker that executes a sequence of jumps in which each individual jump is independent of all prior jumps. Thereby, we deny the ‘walker’ any memory. Such a jump sequence is sometimes denoted as a Markov sequence (memory free walk) or as an uncorrelated random walk. The double sum in Eq. (4.20) contains n(n − 1)/2 average values of the products x i x j  or r i r j .These terms contain memory effects also denoted as correlation effects. For a Markov sequence these average values are zero, as for every pair x i x j one can find for another particle of the ensemble a pair x i x j equal and opposite in sign. Thus, we get from Eq. (4.20) for a random walk without correlation R 2 random  = n  i=1 r 2 i , X 2 random  = n  i=1 x 2 i . (4.21) The index ‘random’ is used to indicate that a true random walk is considered with no correlation between jumps. In a crystal lattice the jump vectors can only take a few definite values. For example, in a coordination lattice (coordination number Z), in which nearest- neighbour jumps occur (jump length d with x-projection d x ), Eq. (4.21) re- duces to  R 2 random  =  n  d 2 ,  X 2 random  =  n  d 2 x . (4.22) Here n denotes the average number of jumps of a particle. It is useful to introduce the jump rate Γ of an atom into one of its Z neighbouring sites via Γ ≡ n Zt . (4.23) We then get D = 1 6 d 2 ZΓ = d 2 6¯τ . (4.24) 62 4 Random Walk Theory and Atomic Jump Process Table 4.1. Geometrical properties of cubic Bravais lattices with lattice parameter a Lattice Coordination number Z Jump length d Primitive cubic 6 a Body-centered cubic (bcc) 8 a √ 3/2 Face-centered cubic (fcc) 12 a √ 2/2 This equation describes diffusion of interstitial atoms in a dilute interstitial solid solution 1 .Thequantity ¯τ = 1 ZΓ (4.25) is the mean residence time of an atom on a certain site. For cubic Bravais lattices, the jump length d and the lattice parameter a are related to each other as indicated in Table 4.1. Using these parameters we get from Eq. (4.24) D = a 2 Γ. (4.26) 4.1.4 Correlation Factor Random walk theory, to this point, involved a series of independent jumps, each occurring without any memory of the previous jumps. However, several atomic mechanisms of diffusion in crystals entail diffusive motions of atoms which are not free of memory effects. Let us for example consider the vacancy mechanism (see also Chap. 6). If vacancies exchange sites with atoms a mem- ory effect is necessarily involved. Upon exchange, vacancy and ‘tagged’ atom (tracer) move in opposite directions. Immediately after the exchange the va- cancy is for a while available next to the tracer atom, thus increasing the probability for a reverse jump of the tracer. Consequently, the tracer atom does not diffuse as far as expected for a completely random series of jumps. This reduces the efficiency of a tracer walk in the presence of positional mem- ory effects with respect to an uncorrelated random walk. Bardeen and Herring in 1951 [7, 8] recognised that this can be ac- counted for by introducing the correlation factor f = lim n→∞ R 2  R 2 random  = 1 + 2 lim n→∞ n−1  i=1 n  j=i+1 r i r j  n  i=1 r 2 i  , 1 In a non-dilute interstitial solution correlation effects can occur, because some of the neighbouring sites are not available for a jump. 4.1 Random Walk and Diffusion 63 f x = lim n→∞ X 2  X 2 random  = 1 + 2 lim n→∞ n−1  i=1 n  j=i+1 x i x j  n  i=1 x 2 i  . (4.27) The correlation factor in Eq. (4.27) equals the sum of two terms: (i) the leading term +1, associated with uncorrelated (Markovian) jump sequences and (ii) the double summation contains the correlation between jumps. It has been argued above that for an uncorrelated walk the double summation is zero. Diffusion in solids is often defect-mediated. Then, successive jumps occur with higher probability in the reverse direction and the contribution of the double sum is negative. Equation (4.27) also shows that one may define the correlation factor as the ratio of the diffusivity of tagged atoms, D ∗ ,and a hypothetical diffusivity arising from uncorrelated jump sequencies, D random , via f ≡ D ∗ D random . (4.28) Correlation effects are important in solid-state diffusion of crystalline mate- rials, whenever diffusion is mediated by a diffusion vehicle (see Table 4.2). Examples for diffusion vehicles are vacancies, vacancy pairs, self-interstitials, etc. An equivalent statement is to say, there must be at least three identi- fiable ‘species’ involved in the diffusion process. For example, during tracer diffusion via vacancies in pure crystals, the three participating ‘species’ are vacancies, host atoms, and tracer atoms. Interstitial diffusion in a dilute in- terstitial solution is uncorrelated, because no diffusion vehicle is involved. Let us consider once more diffusion in a cubic Bravais lattice. When cor- relation occurs, Eq. (4.24) must be replaced by D ∗ = 1 6 fd 2 ZΓ = fa 2 Γ. (4.29) We will return to the correlation factor in Chap. 7, after having introduced point defects in Chap. 5 and the major mechanisms of diffusion in crystals in Chap. 6. Table 4.2. Correlation effects of diffusion for crystalline materials f = 1 Markovian jump sequence No diffusion vehicle involved: direct interstitial diffusion f<1 Non-Markovian jump sequence Diffusion vehicle involved: vacancy, divacancy, self-interstitial, . . . mechanisms 64 4 Random Walk Theory and Atomic Jump Process 4.2 Atomic Jump Process In preceding sections, we have considered many atomic jumps on a lat- tice. Equally important are the rates at which jumps occur. Let us take a closer look to the atomic jump process illustrated in Fig. 4.3. An atom moves into a neighbouring site, which could be either a neighbouring va- cancy or an interstitial site. Clearly, the jumping atom has to squeeze be- tween intervening lattice atoms – a process which requires energy. The en- ergy necessary to promote the jump is usually large with respect to the thermal energy k B T . At finite temperatures, atoms in a crystal oscillate around their equilibrium positions. Usually, these oscillations are not vio- lent enough to overcome the barrier and the atom will turn back to its ini- tial position. Occasionally, large displacements result in a successful jump of the diffusing atom. These activation events are infrequent relative to the frequencies of the lattice vibrations, which are characterised by the De- bye frequency. Typical values of the Debye frequency lie between 10 12 and 10 13 s −1 . Once an atom has moved as the result of an activation event, the energy flows away from this atom relatively quickly. The atom becomes de- activated and waits on the average for many lattice vibrations before it jumps again. Thermally activated motion of atoms in a crystal occurs in a series of discrete jumps from one lattice site (or interstitial site) to the next. The theory of the rate at which atoms move from one site to a neighbour- ing one was proposed by Wert [9] and has been refined by Vineyard [10]. Vineyard’s approach is based upon the canonical ensemble of statistical me- chanics for the distribution of atomic positions and velocities. The jump process can be viewed as occurring in an energy landscape characterised by Fig. 4.3. Atomic jump process in a crystalline solid: the black atom moves from an initial configuration (left) to a final configuration (right) pushing through a saddle- point configuration (middle) 4.2 Atomic Jump Process 65 the difference in Gibbs free energy G M between the saddle-point barrier and the equilibrium position (Fig. 4.3). G M is denoted as the Gibbs free energy of migration (superscript M) of the atom. It can be separated according to G M = H M − TS M , (4.30) where H M denotes the enthalpy of migration and S M the entropy of migra- tion. Using statistical thermodynamics, Vineyard [10] has shown that the jump rate ω (number of jumps per unit time to a particular neighbouring site) can be written as ω = ν 0 exp  − G M k B T  = ν 0 exp  S M k B  exp  − H M k B T  , (4.31) where ν 0 is called the attempt frequency. It is of the order of the Debye frequency. Its rigorous meaning is as follows: ν 0 is the vibration frequency around the equilibrium site but in the direction of the reaction path. The entropy of migration corresponds to the change in lattice vibrations associ- ated with the displacement of the jumping atom from its equilibrium to the saddle point configuration. Vineyard’s treatment also provides an expression for the migration entropy: S M = k B ⎡ ⎣ 3N−1  j=0 ln  hν j k B T  A − 3N−1  j=0 ln  hν  j k B T  SP ⎤ ⎦ . (4.32) The ν j are the 3N − 1 normal mode frequencies for vibrations around the equilibrium site A while ν  j are the frequencies of the system when it is con- strained to move within the hyperface which passes through the saddle point (SP) perpendicular to the jump direction. The theory of the rates at which atoms move from one lattice site to another covers one of the fundamental aspects of diffusion. For most practical purposes, a few general conclusions are important: 1. Atomic migration in a solid is the result of a sequence of localised jumps from one site to another. 2. Atomic jumps in crystals usually occur from one site to a nearest- neighbour site. Molecular dynamic simulations mostly confirm this view. Multiple hops are rare, although their occurrence is indicated in some model substances. Jumps with magnitudes larger than the nearest- neighbour distance are more common on surfaces or in grain boundaries (see Chap. 32). 3. The jump rate ω has an Arrhenius-type dependence on temperature as indicatedinEq.(4.31). 4. The concept of an atomic jump developed above applies to all the diffu- sion mechanisms discussed in Chap. 6. Of course, the values of G M ,H M , 66 4 Random Walk Theory and Atomic Jump Process and S M depend on the diffusion mechanism and on the material under consideration: For interstitial diffusion, the quantities G M , H M ,andS M pertain to the saddle point separating two interstitial positions. For a dilute interstitial solution, virtually every interstitial solute is surrounded by empty inter- stitial sites. Thus, for an atom executing a jump the probability to find an empty site next to its starting position is practically unity. For vacancy-mediated diffusion, G M , H M ,andS M pertain to the saddle- point separating the vacant lattice site and the jumping atom on its equilibrium site. 5. There are cases – mostly motion of hydrogen in solids at low tempera- tures – where a classical treatment is not adequate [16]. However, the end result of different theories including quantum effects is still a movement in a series of distinct jumps from one site to another [17]. For atoms heavier than hydrogen and its isotopes, quantum effects can usually be disregarded. For more detailed discussions of the problem of thermally activated jumps the reader may consult the textbook of Flynn [3] and the reviews by Franklin [11], Bennett [12], Jacucci [13], H ¨ anngi et al. [15] Pon- tikis [14] and Flynn and Stoneham [17]. Molecular dynamic calculations become increasingly important and are used to check and to supplement analytical theories. Atomistic computer simulations of diffusion processes have been reviewed, e.g., by Mishin [18]. Nowadays, simulation methods present a powerful approach to gain funda- mental insight into atomic jump processes in materials. The capabilities of simulations have drastically improved due to the development of new simula- tion methods reinforced by increased computer power. Reliable potentials of atomic interaction have been developed, which allow a quantitative descrip- tion of point defect properties. Simulation of atomic jump processes have been applied to ordered intermetallic compounds, surface diffusion, grain- boundary diffusion, and other systems. The challenge is to understand, de- scribe, and calculate diffusion coefficients in a particular metal, alloy, or com- pound. In some – but not all – of the rare cases, when this has been done, the agreement with experiments is encouraging. References 1. A. Einstein, Annalen der Physik 17, 549 (1905) 2. M. van Smoluchowski, Annalen der Physik 21, 756 (1906) 3. M. van Smoluchowski, Z. Phys. 13, 1069 (1912); and Physikalische Zeitschrift 17, 557 (1916) 4. C.P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, 1972 5. P. H¨anggi, P. Talkner, M. Borkovec: Rev. Mod. Phys. 62, 251 (1990) References 67 6. J.R. Manning, Diffusion Kinetics for Atoms in Crystals, D. van Norstrand Comp., Inc., 1968 7. J. Bardeen, C. Herring, in: Atom Movements, A.S.M. Cleveland, 1951, p. 87 8. J. Bardeen, C. Herring, in: Imperfections in Nearly Perfect Solids, W. Shockley (Ed.), Wiley, New York, 1952, p. 262 9. C. Wert, Phys. Rev. 79, 601 (1950) 10. G. Vineyard, J. Phys. Chem. Sol. 3, 121 (1957) 11. W.M. Franklin, in: Diffusion in Solids – Recent Developments, A.S. Nowick, J.J. Burton (Eds.), Academic Press, Inc., 1975, p.1 12. C.H. Bennett, in: Diffusion in Solids – Recent Developments, A.S. Nowick, J.J. Burton (Eds.), Academic Press, Inc., 1975, p.74 13. G. Jacucci, in: Diffusion in Crystalline Solids,G.E.Murch,A.S.Nowick(Eds.), Academic Press, Inc., 1984, p.431 14. V. Pontikis, Thermally Activated Processes,in:Diffusion in Materials, A.L.Laskar,J.L.Bocqut,G.Brebec,C.Monty(Eds.),KluwerAcademicPub- lishers, Dordrecht, The Netherlands, 1990, p.37 15. P. H¨anngi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990) 16. J. V¨olkl, G. Alefeld, in: Diffusion in Solids – Recent Developments, A.S. Nowick, J.J. Burton (Eds.), Academic Press, Inc., 1975 17. C.P. Flynn, A.M. Stoneham, Phys. Rev. B1, 3966 (1970) 18. Y. Mishin, Atomistic Computer Simulation of Diffusion,in:Diffusion Processes in Advanced Technological Materials, D. Gupta (Ed.), William Andrews, Inc., 2005 5 Point Defects in Crystals The Russian scientist Frenkel in 1926 [1] was the first author to introduce the concept of point defects (see Chap. 1). He suggested that thermal agita- tion causes transitions of atoms from their normal lattice sites into interstitial positions leaving behind lattice vacancies. This type of disorder is nowadays denoted as Frenkel disorder and contained already the concepts of vacancies and self-interstitials. Already in the early 1930s Wagner and Schottky [2] treated a fairly general case of disorder in binary AB compounds considering the occurrence of vacancies, self-interstitials, and of antisite defects on both sublattices. Point defects are important for diffusion processes in crystalline solids. This statement mainly derives from two features: one is the ability of point defects to move through the crystal and to act as ‘vehicles for diffusion’ of atoms; another is their presence at thermal equilibrium. Of particular interest in this chapter are diffusion-relevant point defects, i.e. defects which are present in appreciable thermal concentrations. In a defect-free crystal, mass and charge density have the periodicity of the lattice. The creation of a point defect disturbs this periodicity. In metals, the conduction electrons lead to an efficient electronic screening of defects. As a consequence, point defects in metals appear uncharged. In ionic crystals, the formation of a point defect, e.g., a vacancy in one sublattice disturbs the charge neutrality. Charge-preserving defect populations in ionic crystals in- clude Frenkel disorder and Schottky disorder, both of which guarantee global charge neutrality. Frenkel disorder implies the formation of equal numbers of vacancies and self-interstitials in one sublattice. Schottky disorder con- sists of corresponding numbers of vacancies in the sublattices of cations and anions. For example, in AB compounds like NaCl composed of cations and anions with equal charges opposite in sign the number of vacancies in both sublattices must be equal to preserve charge neutrality. Point defects in semi- conductors introduce electronic energy levels within the band gap and thus can occur in neutral or ionised states, depending on the position of the Fermi level. In what follows, we consider at first point defects in metals and then proceed to ionic crystals and semiconductors. Nowadays, there is an enormous body of knowledge about point defects from both theoretical and experimental investigations. In this chapter, we 70 5 Point Defects in Crystals provide a brief survey of some features relevant for diffusion. For more com- prehensive accounts of the field of point defects in crystals, we refer to the textbooks of Flynn [3], Stoneham [4], Agullo-Lopez, Catlow and Townsend [5], to a review on defect in metals by Wollenberger [6], and to several conference proceedings [9–12]. For a compilation of data on point de- fects properties in metals, we refer to a volume edited by Ullmeier [13]. For semiconductors, data have been assembled by Schulz [14], Stolwijk [15], Stolwijk and Bracht [16], and Bracht and Stolwijk [17]. Proper- ties of point defects in ionic crystals can be found in reviews by Barr and Lidiard [7] and Fuller [8] and in the chapters of Beni ` ere [18] and Erdely [19] of a data collection edited by Beke. 5.1 Pure Metals 5.1.1 Vacancies Statistical thermodynamics is a convenient tool to deduce the concentra- tion of lattice vacancies at thermal equilibrium. Let us consider an elemental crystal, which consists of N atoms (Fig. 5.1). We restrict the discussion to metallic elements or to noble gas solids in which the vacancies are in a single electronic state and we suppose (in this subsection) that the concentration is so low that interactions among them can be neglected. At a finite temper- ature, n 1V vacant lattice sites (monovacancies, index 1V ) are formed. The total number of lattice sites then is N  = N + n 1V . (5.1) The thermodynamic reason for the occurrence of vacancies is that the Gibbs free energy of the crystal is lowered. The Gibbs free energy G(p, T )ofthe Fig. 5.1. Vacancies in an elemental crystal 5.1 Pure Metals 71 crystal at temperature T and pressure p is composed of the Gibbs function of the perfect crystal, G 0 (p, T ), plus the change in the Gibbs function on forming the actual crystal, ∆G: G(p, T )=G 0 (p, T )+∆G, (5.2) where ∆G = n 1V G F 1V − TS conf . (5.3) In Eq. (5.3) the quantity G F 1V represents the Gibbs free energy of formation of an isolated vacancy. It corresponds to the work required to create a va- cancy by removing an atom from a particular, but arbitrary, lattice site and incorporating it at a surface site (‘Halbkristalllage’). Not only surfaces also grain boundaries and dislocations can act as sources or sinks for vacancies. If a vacancy is created, the crystal lattice relaxes around the vacant site and the vibrations of the crystal are also altered. The Gibbs free energy can be decomposed according to G F 1V = H F 1V − TS F 1V (5.4) into the formation enthalpy H F 1V and the formation entropy S F 1V .Thelast term on the right-hand side of Eq. (5.3) contains the configurational entropy S conf , which is the thermodynamic reason for the presence of vacancies. In the absence of interactions, all distinct configurations of n 1V vacancies on N  lattice sites have the same energy. The configurational entropy can be expressed through the equation of Boltzmann S conf = k B ln W 1V , (5.5) where W 1V is the number of distinguishable ways of distributing n 1V mono- vacancies among the N  lattice sites. Combinatoric rules tell us that W 1V = N  ! n 1V !N! . (5.6) The numbers appearing in Eq. (5.6) are very large. Then, the formula of Stirling, ln x! ≈ x ln x, approximates the factorial terms and we get ln W 1V ≈ (N + n 1V )ln(N + n 1V ) −n 1V ln n 1V − N ln N. (5.7) Thermodynamic equilibrium is imposed on a system at given temperature and pressure by minimising its Gibbs free energy. In the present case, this means ∆G ⇒ Min . (5.8) The equilibrium number of monovacancies, n eq 1V , is obtained, when the Gibbs free energy in Eq. (5.3) is minimised with respect to n 1V , subject to the constraint that the number of atoms, N, is fixed. Inserting Eqs. (5.5) and (5.7) [...]... about 16 0 ps near room temperature and reaches a high Fig 5.5 Mean lifetime of positrons in Al according to Schaefer et al [28] 5 .1 Pure Metals 79 eq Table 5 .1 Monovacancy properties of some metals C1V is given in site fractions Metal F H1V /eV F S1V /kB eq C1V at Tm Method(s) Al Cu Au Ag Pb Pt Ni Mo W 0.66 1. 17 0. 94 1. 09 0 .49 1. 49 1. 7 3.0 4. 0 0.6 1. 5 1. 1 – 0.7 1. 3 – – 2.3 9 .4 × 10 4 2 × 10 4 7.2 × 10 4. .. ionic crystals Ionic compound (HSP or HFP )/eV (SSP or SFP )/kB Type of disorder NaCl KCl NaI KBr LiF LiCl LiBr LiI 2 .44 2. 54 2.00 2.53 2.68 2 .12 1. 80 1. 34 9.8 7.6 7.6 10 .3 Schottky Schottky Schottky Schottky Schottky Schottky Schottky Schottky AgCl AgBr 1. 45 1. 55 1. 13 1. 28 5 .4 12 .2 6.6 12 .2 Frenkel Frenkel This type of disorder is called Schottky disorder and KSP is denoted as the Schottky product Charge... equilibrium, ∂∆G/∂n1V = 0: F F H1V − T S1V + kB T ln neq 1V =0 N + neq 1V (5.9) By definition we introduce the site fraction of monovacancies 1 via: C1V ≡ n1V N + n1V (5 .10 ) This quantity also represents the probability to find a vacancy on an arbitrary, eq but particular lattice site In thermal equilibrium we have C1V ≡ neq /(N + 1V eq n1V ) Solving Eq (5.9) for the equilibrium site fraction yields eq C1V = exp... eq C1V = exp − GF 1V kB T = exp F S1V kB exp − F H1V kB T (5 .11 ) This equation shows that the concentration of thermal vacancies increases via a Boltzmann factor with increasing temperature The temperature depeneq dence of C1V is primarily due to the formation enthalpy term in Eq (5 .11 ) We note that the vacancy formation enthalpy is also given by F H1V = −kB eq ∂ ln C1V ∂ (1/ T ) (5 .12 ) This quantity... the Gibbs free energy 1V of vacancy formation next to a B atom, GF (B): 1V GF (A) = GF (B) 1V 1V (5.27) For GF (A) > GF (B) the vacancy-solute interaction is attractive, whereas 1V 1V for GF (A) < GF (B) it is repulsive According to Lomer [32] the total 1V 1V eq vacancy fraction in a dilute alloy, CV (CB ), is given by eq CV (CB ) = (1 − ZCB ) exp − GF (A) GF (B) 1V + ZCB exp − 1V , kB T kB T Fig 5.7... × 10 4 7.2 × 10 4 1. 7 × 10 4 1. 7 × 10 4 – – – 1 × 10 4 DD + PAS DD + PAS DD DD DD RQ PAS PAS RQ + TEM temperature value of about 250 ps From a fit of Eq (5.23) to the data the product σC1V can be deduced If the trapping cross section is known the vacancy concentration is accessible If σ is constant, the vacancy formation enthalpy can be deduced from the temperature variation of σC1V At high temperature,... crystal may end its existence as a ‘free’ particle either by the annihilation rate 1/ τf or by being trapped by a vacancy with the trapping rate σC1V , where σ is the trapping cross section This results in a lifetime given by τf / (1 + τf σC1V ) If one assumes that initially all positrons are free, one gets for their mean lifetime: τ = τf ¯ 1 + τt σC1V 1 + τf σC1V (5.23) Figure 5.5 shows as an example... fast ionic conduction was reported as early as 19 14 [35] It displays a first order phase transition between a fast ionconducting phase (α-AgI) above 14 7 ◦ C and a normal conducting phase at lower temperatures α-AgI has a body-centered cubic sublattice of practically immobile I− ions Each unit cell displays 42 interstitial sites (6 octahedral, 12 tetrahedral, 24 trigonal) over which the two Ag+ ions per... which according to 2V B B GB = H2V − T S2V 2V (5 . 14 ) B B can be decomposed into an enthalpy H2V and an entropy S2V of interaction B For G2V > 0 the interaction is attractive and binding occurs, whereas for GB < 0 it is repulsive Combining Eq (5 . 14 ) with the mass-action law for 2V Eq (5 .13 ) yields GB Z eq eq 2V (5 .15 ) C2V = exp (C1V )2 2 kB T Equation (5 .15 ) shows that at thermal equilibrium the divacancy... agglomerates) is then eq eq eq CV = C1V + 2C2V (5 .16 ) For a typical monovacancy site fraction in metals of 10 4 near the melting temperature (see below), the fraction of non-interacting divacancies would be Fig 5.2 Arrhenius diagram of equilibrium concentrations of mono- and divacancies in metals (schematic) 74 5 Point Defects in Crystals × 10 −8 Typical interaction energies of a few 0 .1 eV increase the divacancy . C eq 1V is given in site fractions Metal H F 1V /eV S F 1V /k B C eq 1V at T m Method(s) Al 0.66 0.6 9 .4 10 4 DD + PAS Cu 1. 17 1. 5 2 10 4 DD + PAS Au 0. 94 1. 1 7.2 × 10 4 DD Ag 1. 09 – 1. 7 10 4 DD Pb. 10 4 DD Pb 0 .49 0.7 1. 7 10 4 DD Pt 1. 49 1. 3 – RQ Ni 1. 7 – – PAS Mo 3.0 – – PAS W4.02. 31 10 4 RQ+TEM temperature value of about 250 ps. From a fit of Eq. (5.23) to the data the product σC 1V can. der Physik 17 , 549 (19 05) 2. M. van Smoluchowski, Annalen der Physik 21, 756 (19 06) 3. M. van Smoluchowski, Z. Phys. 13 , 10 69 (19 12); and Physikalische Zeitschrift 17 , 557 (19 16) 4. C.P. Flynn,