The Clausius-Mossotti Equation

Transcription

1 The Clausius-Mossotti quation Ottaviano Mossotti (1850) and Rudolf Clausius (1879) Jason Rich McKinley Goup Summe Reading Club Familia Results and Famous Papes August 17, 2007

2 Ottaviano Fabizio Mossotti ( ) Italian physicist and mathematician Completed studies at Univesity of Pavia in 1811 at age 20 Contibutions in: Astophysics (Bea Obsevatoy, ) Molecula physics Inteesting facts: Pat of a secet society against the occupying Austian govenment scaped to London in 182, then to Buenos Aies, Agentina, until 185 Retuned to Italy in 1840, whee he taught until his death 2

3 Rudolf Julius mmanuel Clausius ( ) Famous quotes: Geman themodynamicist extaodinaie Doctoal thesis on atmospheic optics fom Univesity of Halle in 1847 Taught in Belin, Wüzbug, Bonn, and biefly at TH Züich A founding fathe of themodynamics Revised the fist and second laws of themo Mathematically descibed entopy and coined the tem The enegy of the univese is constant The entopy of the univese tends to a maximum Clausius-Clapeyon quation ( P) ( 1 ) [ L V] dln ΔH = d T R vap Teste, J.W., Modell, M. (1997), Themodynamics and Its Applications, d d., Pentice Hall, New Jesey, p. 681.

4 The quation ε 1 χ Nα = = ε + 2 χ + ε ε 1 χ 4 = = π NαV ε + 2 χ + 0 ε = bulk dielectic constant 2 4 A s ε 0 = dielectic pemittivity of fee space χ = m kg bulk dielectic susceptibility 2 4 A s α = molecula polaizability kg αv = molecula polaizability volume ( m ) N = Numbe of dipola molecules pe unit volume Relates bulk, macoscopic quantity (ε o χ) to molecula quantity (α) Deived independently (supposedly) by Mossotti (1850) and Clausius (1879) Clausius, R. (1879), Die Mechanische Wämetheoie, 2, Baunschweig, p Mossotti, O.F. Mem. Di Math, e Fisica d. Soc. Italiana d. Scienze, 24, 2, (1850), 49. 4

5 What s the big deal? Useful Claifies molecula oigin of dielectic constant Allows calculation of molecula polaizability fom measuements of ε ε 1 χ Nα = = ε + 2 χ + ε 0 Deivation is ticky, yet applicable to many physical situations: What is the local electic field of a dielectic on a molecula scale? ext p loc ext loc =? 5

6 Deivation Path ext polaizes dielectic Relate ext to loc Plug loc into eqn: P= Np = Nε α = ε χ i i i 0 loc 0 Solve fo χ Relatively long ange inteactions between dipoles (U ~ 1/ ) loc Key Assumptions: a) Only dipola inteactions Split mateial into discete and continuum egions b) All dipole moments ae identical c) Molecules distibuted isotopically d) Fluctuations negligible 6

7 Claification and Coection of qn 1.2 lectic field fom a polaized body: n n (,, ) = p i p n = dipole moment of molecule n P ( ξηζ,, ) φ n xyz p n 0 n (ξ,η,ζ) dielectic z y x ( ξηζ,, ) dφ = P i dv φ dξdηdζ = P i 0 V 0 P S ip 1 Si = + Pi S = e + e + e ( x ξ) ( y η) ( z ζ ) n x y z (x,y,z) ( ) R dv = n= 1 M = p n ( ) ip R d body 0R12 Define two diffeent gadient F = ex + ey + e = e + e + e z opeatos fo the two coodinates: x y z ξ η ζ 1 We now ecognize: S = 1 1 φ ( xyz,, ) = S dv Pi 0 The dot poduct is expanded: S x y z ( xyz,, ) V Then use the divegence theoem to get: PidS ( S ip) dv φ ( xyz,, ) = + S 0 V 0 ( ip) PidS S dv = F S V 7 R

8 Finding the Local Field, loc Adopted fom Fölich, Theoy of Dielectics, Conside a macoscopic dielectic in an extenal field e+ e- ext a 0 Conside a cubic lattice of dipoles Assumptions: Chage sepaation << a 0 Popeties in sphee ae same as bulk, fluctuations negligible Only dipola inteactions 8

9 Finding the Local Field, loc Assume that applying polaizes each lattice site the same amount: loc e- e+ Restoing foce ~ balances the foce fom loc 2 2 e Restoing foce = c = eloc p= e = c loc Sepaate contibutions of loc into inside and outside of sphee: loc = in + out Find in by summing each dipole inteaction in the sphee: 2 p 2 Inteaction enegy between 2 identical point dipoles = Uij = ( 1 cos ( θij )) lij l = Distance between dipoles i and j ij Since dipoles ae in lattice, l ij = (ma 0, na 0, qa 0 ) whee m, n, q ae integes a 0 ext e+ e- U p m + n 2q = U = = 0 ij ( + + ) a0 mnq,, m n q 9 52 So in = 0!

10 Finding the Local Field, loc loc = in + out Calculate out macoscopically ext s So finally: 4π loc = in + out = S = + P ε + 2 loc = Loentz fomula Plug into loc the micoscopic equation 2 e ε + 2 ε 1 P= N = c 4π P out = s s = self-field = field at cente of a pemanently polaized sphee - Can be found with a standad calculation 4π S = P Hee the macoscopic ε 1 P= elation between P and is: 4π 2 And the micoscopic 1 e p= P= elation is: N c ε 1 4π e 4 = N = π NαV ε + 2 c 2 10 loc

11 Limitations of the quation Condensed systems (high density) van de Waals and multipole foces can become significant If we eaange the C-M eqn, we get: 8π NαV ε = + Citical density at N 4π NαV Not obseved in = πα 4 V expeiments Systems of pemanently pola molecules The deivation assumes that all polaity is induced Pemanent dipoles equie a coection to the local field, as will be seen Böttche, C. J. F. (1952), Theoy of lectic Polaisation, lsevie Publishing Co., New Yok, p

12 Onsage s Coection to Local Field When calculating loc, we neglected the coupling of the cente dipole with the suounding ones e+ e- ext a 0 Onsage coected this: co = loc - eaction e+ e- a = 2χ p + eaction 2χ 0a This yields: 2 χ 1+ χ Goes to C-M eqn Nα in low density limit = 1+ χ χ = Nα N + N α 12 4

13 Altenative Deivations Abound Hannay gives a deivation which involves no splitting into inne and oute pats Uses the full expession fo field of a dipole: ( i ) p p 4π = p δ Local field coection esults natually Othe deivations based on quantum mechanics Attempt to account fo non-localization of electons at lattice points and othe moe complex phenomena g, Adle, Onodea 5 ( ) S. L. Adle, Phys. Rev. 126 (1962) 41. J.H. Hannay u. J. Phys. 4 (198) 141. Y. Onodea, Pog. Theo. Phys., 49 (197) 7. 1

14 Applications in Othe Aeas Dispesion popeties in optical fibes (P. Melman and R. W. Davies J. Lightwave Tech. (1985) 112. Relating mola efactivity to bulk efactive index (Boling, Glass, Owyoung, I J. Quant. lectonics 14 (1978) 601.) 14

15 Thank you fo you attention! ε 1 χ 4 = = π NαV ε + 2 χ + ε 1 χ Nα = = ε + 2 χ + ε 0 15