3.225 17 • Usually Clausius-Mosotti necessary due to high density of dipoles Ionic Polarizability ( − ++== + − −+ 22 2 3 1 32 1 ωω αα εε α ε ε oioo tot r r M e v N By convention, things are abbreviated by using ε s and ε ∞ : ( ++= + − << −+ 2 2 3 1 2 1 , oios s oi M e v ω αα εε ε ωω [ −+ ∞ ∞ += + − = + − >> αα εε ε ωω vn n o oi 3 1 2 1 2 1 , 2 2 + + = − − += ∴ ∞∞ ∞ 2 2 , 1 22 2 2 s oiT T s r ε ε ωω ω ω εε εε ε r ω ω T n 2 =ε ∞ ε s © E. Fitzgerald-1999 ) ) ] 3.225 18 Orientational Polarizability • No restoring force: analogous to conductivity H H O p + - C O O p=0 +q -q θ For a group of many molecules at some temperature: Tk pE Tk U bb eef θ cos == − After averaging over the polarization of the ensemble molecules (valid for low E-fields): Tk p b DC 3 ~ 2 α Analogous to conductivity, the molecules collide after a certain time t, giving: ωτ α α i DC o − = 1 © E. Fitzgerald-1999 9 10 3.225 19 Dielectric Loss © E. Fitzgerald-1999 • For convenience, imagine a low density of molecules in the gas phase • C-M can be ignored for simplicity • There will be only electronic and orientational polarizability ωτ ε ε ε α εεωτ ωτε α χχε i n n N n i N n so r o DC sor o DC oer − − += ∴ +==<< − +=++= 1 3 , ,1 )1(3 1 2 2 2 2 We can write this in terms of a real and imaginary dielectric constant if we choose: ωτ τω ε ε τω ε ε εεε 22 2 so 22 2 2 1 '' ; 1 ' ''' + − = + − += += nn n i so r Water molecule: τ=9.5x10 -11 sec, ω~10 10 microwave oven, transmission of E-M waves logωτ -2 0 +2 ε’,ε’’ n 2 ε so α e +α i α e 3.225 20 Dielectric Constant vs. Frequency • Completely general ε due to the localized charge in materials ω ε 1/τ ω T ω oe 1 n 2 α o α i α e molecules ions electrons Dispersion-free regions, v g =v p © E. Fitzgerald-1999 11 3.225 21 Dispersion • Dispersion can be defined a couple of ways (same, just different way) – when the group velocity ceases to be equal to the phase velocity – when the dielectric constant has a frequency dependence (i.e. when dε/dω not 0) k ω Dispersion-free Dispersion k c r ε ω = g r p v k c k v = ∂ ∂ === ω ε ω g r p v k c k v = ∂ ∂ ≠== ω ωε ω )( © E. Fitzgerald-1999 1 3.225 1 Spontaneous Polarization Remember form of orientational polarization: kT C kT p or == 3 2 α With C ≡ Curie constant Define a critical temperature T c by k NC T c 0 3 ε = Noting further Thus © H.L. Tuller, 2001 or 0 3 ε α orc N T T = Fig. 1. The Curie-Weiss law illustrated for (Ba,Sr)TiO 3 From L.L. Hench and J.K. West, Principles of Electronic Ceramics, Wiley, 1990, p. 243. 1 33 00 = = c kT CNN εε α c c TT T − = 3 χ 3.225 2 • Each unit cell a dipole! •Large P R (remnant polarization, P(E=0) • Coercive Field E C , electric field required to bring P back to zero. Ferroelectrics E R R o ∆E Two equivalent-energy atom positions Can flip cell polarization by applying large enough reverse E-field to get over barrier E P ‘normal’ dielectric P s E c P R © E. Fitzgerald-1999 3.225 3 Ferroelectrics • ‘Confused’ atom structure creates metastable relative positions of positive and negative ions © E. Fitzgerald-1999 3.225 4 Ferroelectrics Applications • Capacitors • Non-volatile memories • Photorefractive materials © H.L. Tuller, 2001 2 3 3.225 5 Characteristics of Optical Fiber •Snell’s Law n 1 n 2 θ 1 θ 2 Refraction Boundary conditions for E-M wave gives Snell’s Law: 2211 sinsin θθ nn = n 2 n 1 θ 1 θ 2 Internal Reflection: θ 1 =90° 2 1 1 2 sin n n c − ==θθ Glass/air, θ c =42° © E. Fitzgerald-1999 3.225 6 • Attenuation – Absorption • OH- dominant, SiO 2 tetrahedral mode – Scattering • Raleigh scattering (density fluctuations) α R ~const./λ 4 (<0.8 µm not very useful!) •Dispersion – material dispersion (see slide i13) – modal dispersion Characteristics of Optical Fiber x • Light source always has ∆λ • parts of pulse with different l propagate at different speeds Black wave arrives later than red wave Solution: grade index y n n 2 n 1 © E. Fitzgerald-1999 3.225 7 Characteristics of Optical Fiber © E. Fitzgerald-1999 3.225 8 Characteristics of Optical Fiber © E. Fitzgerald-1999 4 3.225 9 Colors Produced by Chromium Above: alexandrite, emerald, and ruby. Center: carbonate, chloride, oxide. Below: potassium chromate and ammonium dichromate. © H.L. Tuller, 2001 3.225 10 Electron distribution in the ground state of a chromium atom (A) and a trivalent chromium ion (B). Chromium Electronic Structure © H.L. Tuller, 2001 5 3.225 11 Interaction of the d orbitals of a central ion with six ligands in an octahedral arrangement. Octahedral Environment of Transition Metal Ion © H.L. Tuller, 2001 3.225 12 The splitting of the five 3d orbitals in a tetrahedral and an octahedral ligand field. Note: hen the element is a mid-gap dopant, transitions within this element lead to absorption and/or emission via luminescence Crystal Field Splitting © H.L. Tuller, 2001 W 6 3.225 13 Optical Transitions in Ruby Optical absorption spectrum tied to Cr transitions in ruby. © H.L. Tuller, 2001 3.225 14 Optical Transitions in Emerald Optical absorption spectrum tied to Cr transitions in emerald. © H.L. Tuller, 2001 7 . index y n n 2 n 1 © E. Fitzgerald-1999 3.225 7 Characteristics of Optical Fiber © E. Fitzgerald-1999 3.225 8 Characteristics of Optical Fiber © E. Fitzgerald-1999 4 3.225 9 Colors Produced. state of a chromium atom (A) and a trivalent chromium ion (B). Chromium Electronic Structure © H.L. Tuller, 2001 5 3.225 11 Interaction of the d orbitals of a central ion with six ligands. metastable relative positions of positive and negative ions © E. Fitzgerald-1999 3.225 4 Ferroelectrics Applications • Capacitors • Non-volatile memories • Photorefractive materials © H.L. Tuller,