Sensitivity of approximate formulas for determining optical constants of thin films

The sensitivity of approximate formulas for determining the optical constants of thin films using measurement of reflectancesR and transmittancesT at normal incidence have been investigated theoretically. The ranges of refractive indexn, absorption indexk,2nk (=₂) andn ²–k ²(=₁) within relative errors of 5%, 10%, and 20% may be obtained. Selected signs of (₁)⁺ or (₁)^– have been determined. Validity of the condition n₀ A=n _s A has been also evaluated (A=1–R –T andA=1–R–T).     

Levinson's Theorem for the Schrödinger Equation in One Dimension

Levinson's theorem for the one-dimensional Schrödinger equation with asymmetric potential which decays at infinity faster thanx ^–2 is established by theSturm-Liouville theorem. The critical case where the Schrödinger equation hasa finite zero-energy solution is also analyzed. It is demonstrated that the numberof bound states with even (odd) parityn ₊(n _–) is related to the phase shift ₊ (0)[_– (0)] of the scattering states with the same parity at zero momentum as ₊ (0)+ /2 =n ₊ and _– (0) =n _– for the noncritical case, and ₊ (0) =n ₊ and_– (0) – /2 =n _– for the critical case.     

The extended phase space of the BRS approach

The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceS _ext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldS _ext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC (S _ext). The action of the BRS operator is analyzed for the caseS=R ²ⁿ with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n–2m) occurs via an intermediate osp(m; 2n–m). Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inS _ext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.     

Intersections of random walks in four dimensions. II

Letf(n) be the probability that the paths of two simple random walks of lengthn starting at the origin in ⁴ have no intersection. It has previously been shown thatf(n)c(logn)^–1/2. Here it is proved that for allr>1/2, .Research Supported by NSF grant MCS-8301037     

The metal-semiconductor transition in amorphous Si1−x Cr x films:T 0.19-contribution to the metallic conductivity

A detailed phenomenological re-analysis of previously published conductivity data, (T, x), is presented. It is based on the investigation of differences, (T, x ₁)–(T, x ₂). In this way, the cusp-like low-temperature term is amplified against the other temperature dependent contributions. This term can be described by wherep=0.19±0.03. It is present, if (4.2 K,x) exceeds 260 ^–1 cm^–1, at least up to (4.2 K,x)1350 ^–1 cm^–1 and forT60 K. But it is absent, if (4.2 K,x)180 ^–1 cm^–1. The disappearance of this contribution should be related to the metal-semiconductor transition, taking place atx _c 0.14. On the other hand, the presence of a term proportional toT ^1/2, as predicted by Altshuler and Aronov, seems unlikely.It is argued that the term should be related to the interplay of electron-electron interaction and disorder. The comparison with data from the literature shows that this contribution might also be present in heavily doped crystalline semiconductors.     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Inverse Problem for a General Bounded Domain inR3 with Piecewise Smooth Mixed BoundaryConditions

We study the influence of a finite container on an ideal gas. The trace of theheat kernel (t) = _{= 1}exp(–t), where {} _{= 1}are the eigenvalues of the negative Laplacian – ² = – ³ _{= 1}(/x )² in the (x ¹, x ², x ³)-space,is studied for a general bounded domain with a smooth bounding surface S, where afinite number of Dirichlet, Neumann, and Robin boundary conditions on thepiecewise smooth parts S _i (i = 1, ..., n) of S are considered such that S =U _{i = 1} S _i . Some geometrical properties of (the volume, the surface area, the meancurvature, and the Gaussian curvature) are determined. Furthermore,thermodynamic quantities, particularly the energy, for an ideal gas enclosed inthe general bounded domain with Dirichlet, Neumann, and Robin conditionsare examined with the help of the asymptotic expansions of (t) for short timet. We show that these thermodynamic quantities depend on some geometricproperties of .     

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].     

The effect of a constant number of particles on the pair correlation function and the density profile in a one-dimensional lattice gas

A one-dimensional lattice gas (Ising model) of lengthL and with nearest-neighbor couplingJ is considered in a canonical ensemble with fixed number of particlesN=L/2. Exact expressions and asymptotic forms for largeL are derived for the density-density correlation function, using periodic boundary conditions, and for the density (magnetization) profile, using antisymmetric boundary conditions. The density-density correlation function,g, assumes for temperaturesT> T, withT = 2J(_BlnL)^–1 and forL large, the formg(x) =g ^gc(x) +BL ^–1 +a(x)L ^–1 +O(L^–2) wherex is a distance between considered lattice sites,B is known from earlier work of Lebowitz and Percus,(1b) anda(x) decays exponentially forx . For TT, the correlation function and the density profile behave differently, the latter exhibiting a step in the middle of the interface.     

Mössbauer study of a novel binuclear Fe(II/III) delocalized-valence compound

In this paper, we briefly summarize the main conclusions of the Mössbauer analysis of [L₂Fe₂(-OH)₃] (ClO₄)₂·2CH₃OH·2H₂O with L=N,N',N"-trimethyl-1,4,7-triazacyclononane, a novel dimeric iron compound, which possesses a central exchange-coupled delocalized-valence Fe(II/III) unit. The complete delocalization of the excess electron in the dimeric iron center is concluded from the indistinguishability of the two iron sites in Mössbauer spectroscopy. The magnetic Mössbauer spectra imply a system spinS _t=9/2 for the dimer in its ground state. The values for hyperfine and spin-Hamiltonian parameters, obtained from simulations of the Mössbauer spectra, are =0.74 mms^–1, E _Q=–2.14 mms^–1,A =–10.6 T,A =–13.5 T andD=1.8 cm^–1. The system spinS _t=9/2 is interpreted to be a consequence of double-exchange coupling.     

Dependency of control parameters on a rate coefficient giving the potential curvature in a reaction cascade model

Many chemical reactions in vivo are self-controlled by fluxes of chemical energy and matter through biological systems, so the induction of such reactions can be governed by changes in the control parameters of the rate equation. A potential of a system is assumed to be given by Gibbs' functionG(T, P, x), which is continuously differentiable, and the rate equation can be derived from the differential (–G/x) of Taylor's expansion ofG _(T,P)(x) for the order parameterx, which corresponds to the product number, at around the critical pointC(T _C, P_C). The equation is described bydx/dt=(x)–k₁x–k₂x³, andk ₂>0. In this equation,k ₁ andk ₂ are functions of the control parameters, temperatureT and pressureP, andk ₁ is allowed to have a positive or negative values as (T, P). Thenk ₁ is an important factor that decides the induction conditions of the reactions with a phase transition in the steady statex=0. Because bothk ₁ (the transition parameter) andG are the quantity of state, they are given by the total differential, and functions that decideG andk ₁ are related to a mutual inverse function. From the above relation, the rate of change ink ₁ by G, which corresponds to the reaction energy of the system, is uniquely determined by a function ofk ₁, [f(k ₁ ^± )] andf(k ₁ ^± ) is described approximately by ±₁ k ₁ ^± in the transient process thatk ₁ approaches zero, where ₁ implies 1/RT. These results indicate that internal driving forces caused by a stimulus in a system are proportional tok ₁ ^± and that the system is regulated by competition of the forces. an approximate function fork ₁ in the transient process is described by tanh (G/RT) and Arrhenius' law is elucidated from this theory.Decreased January 19, 1992     

On the asymptotics of occurrence times of rare events for stochastic spin systems

We consider translation-invariant attractive spin systems. LetT _,x ^v be the first time that the average spin inside the hypercube reaches the valuex when the process is started from an invariant measure with density smaller thanx. We obtain sufficient conditions for (1) ¦¦^–1 logT _,x ^v (x) in distribution as ¦¦ , and ¦¦^–1 logT _,x ^v (x) as ¦¦ , where (x):= –lim ¦¦^–1 log {(average spin inside ) x. And (2)T _,x ^v /ET _,x ^v converges to a unit mean exponential random variable as ¦¦ . Both (1) and (2) are proven under some type of rapid convergence to equilibrium. (1) is also proven without extra conditions for Ising models with ferromagnetic pair interactions evolving according to an attractive reversible dynamics; in this case is a thermodynamic function. We discuss also the case of finite systems with boundary conditions and what can be said about the state of the system at the timeT _,x ^v .On leave from São Paulo University.     

Negative muon precession in nuclei with spin

Polarized negative muons were stopped in various materials containing nuclei with nonzero spin. The TF-SR precession signal of theF ⁺ hyperfine state (frequencyv ₊ > 0) was pronounced for ^–Li and ^–Be, faint for ^–Cl (in NiCl₂), and undetectable for ^–F (in CaF₂ or NiF₂) and ^–P. TheF ^– signal (frequencyv _– < 0) was observed clearly for ^–Be, ^–Al and ^–Na, marginally for ^–K, ^–V and ^–Nb, and not at all for ^–Ga (at 4 kOe) or ^–Co (ferrromagnetic, zero field). In the heavier elements theF ^– signal is fed by transitions from theF ⁺ to theF ^– state at a rateR, as long asR(v ₊ –v _–). [See separate paper on ^–Al in these Proceedings, p. 879.]I am grateful to Alex Schenck and Bruce Patterson for the loan of equipment and samples, to Fred Gygax for helping set up the apparatus, to Jun Imazato for help with the experiment, and to Toshi Yamazaki and Tak Suzuki for valuable comments. I am also indebted to SIN for several weeks of free ^– beam time and to SIN, BOOM, and the University of Tokyo for free time on their VAX computers, which were kept busy for several months in the analysis.     

Electron Energy Loss Spectra of the Organic Complexes of Europium

In the electron energy loss spectra (EELS) of the organic europium complexes Eu³⁺ (BTFA)₃TPPO and Eu³⁺(BrBTFA)₃TPPO in a gas phase obtained on excitation by monokinetic beams of electrons of different energies in the range 12–50 eV, we have identified the bands associated with the electron transitions S ₀–S ₁, S ₀–S ₂, and S ₀–S ₃. The connection of these transitions with the structural groups of the complexes is established. The addition of the bromine atom to the phenyl ring of diketonate leads to the rise in the relative intensity of the S ₀–S ₂ band. The singlettriplet transitions manifest themselves in the region 2.5–3.2 eV and contribute to the S ₀–S ₂ band of the electron energy loss spectra.     

Scaling solutions of Smoluchowski's coagulation equation

We investigate the structure of scaling solutions of Smoluchowski's coagulation equation, of the formc _k (t)s(t)^– (k/s(t)), wherec _k (t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, and(x) is a scaling function. For the rate constantK(i, j) in Smoluchowski's equation, we make the very general assumption thatK(i, j) is a homogeneous function of the cluster sizesi andj:K(i,j)=a ^– K(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j0 asj. We show that gelation occurs if>1, and does not occur if1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation for(x). We present a detailed analysis of the behavior of(x) at large and small values ofx. For all models, we find exponential large-x behavior: (x)A x ^– e ^–x asx and, for different kernelsK(i, j), algebraic or exponential small-x behavior: (x)Bx ^– or (x)=exp(–Cx ^–|| + ...) asx0.     

Fine Structures of the Permittivity and Characteristic Losses of Electrons in Fluorite

Experimental-computational spectra of the permittivity and characteristic losses –Im^–1 for energies in the range 5–21 eV at a temperature of 4.2 K and theoretical spectra of and –Im^–1 of a fluorite crystal are resolved into elementary transition bands. The parameters of transition bands (energies of their maxima E _i, band halfwidths H _i and areas S _i, and oscillator forces f _i) are determined. A correlation of the spectral bands of and –Im^–1is established, and their specific features are elucidated.     

The inverses-wave scattering problem for a class of potentials depending on energy

The inverse scattering problem is considered for the radials-wave Schrödinger equation with the energy-dependent potentialV ⁺(E,x)=U(x)+2 Q(x). (Note that this problem is closely related to the inverse problem for the radials-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the caseQ=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solutionf ⁺(E,x) is shown to be generated by two functionsF ⁺(x) andA ⁺(x,t). After introducing the potentialV ^–(E,x)=U(x)–2 Q(x) and the corresponding functionsF ^–(x) andA ^–(x,t), fundamental integral equations are derived connectingF ⁺(x),F ^–(x),A ⁺(x,t) andA ^–(x,t) with two functionsz ⁺(x) andz ^–(x);z ⁺(x) andz ^–(x) are themselves easily connected with the binding energiesE _n ⁺ and the scattering matrixS ⁺(E),E>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.Physique Mathématique et Théorique, Equipe de recherche associée au C.N.R.S.     

Rates of change in high temperature electrical resistivity and oxygen diffusion coefficient in Ba2YCu3Ox

The change in electrical resistance with time for bulk, thick-film, and thin-film Ba₂YCu₃O_x at atmospheric pressure is described as a function of the oxygen partial pressure (100 to 0.001%) and temperature (320°–750°C). The potential usefulness of these materials as oxygen sensors is demonstrated. The rate of equilibration is faster during oxygen uptake than during its loss. Time constants to reach equilibration (1/e remaining), qualitatively scale with sample dimensions. For a 1m film at 600° C, <1 s for the range of P_O2 (O2 being a shorthand for O₂) from 100% to 0.001%. The rate increases markedly with increasing P_O2. The actual resistance decreases with P_O2 at a rate of log/log P_O2 = 0.4 at 700° C showing adequate sensitivity for sensor purposes. Times for the transient resistance change in the sample where used to estimate the oxygen diffusion coefficient in the ceramic. The diffusivities obtained are 4·10^–11–1·10^–12 cm²/s in the 435°–320° C range, with an activation energy of 27 kcal/mole.     

155Gd Mössbauer investigation on (GdxNd1−x)1+εFe4B4

Tetragonal (Gd_xNd_1–x)₁₊ Fe₄B₄ alloys have been investigated for 0.2x1 by Mössbauer spectroscopy, using the 86.5 keV¹⁵⁵Gd resonance. The Gd quadrupolar interaction e²qQ=12.67(5) mm/s for x=1, nearly independent of x, is the largest observed to date in metallic compounds of Gd. A crystal field term A ₂ ⁰ =–2450±50 K/a ₀ ² is inferred. This quadrupolar interaction shows some dispersion increasing when x decreases, reflecting the quasi incommensurate nature of the (Gd,Nd) and Fe+B sublattices in the (Gd_xNd_1–x)Fe₄B₄ structure (=0.109 for x=0 and =0.139 for x=1). The hyperfine field is perpendicular to the c axis for x0, but no unique direction is obtained for x=0.     

A microscopic calculation of dynamic critical exponents for displacive phase transitions

For anO(n)-isotropic lattice dynamicalQ ⁴-model describing displacive phase transitions ind dimensions, we employ a microscopic 1/n-expansion in order to show that over-damped soft-phonon behavior emerges for frequencies smaller than those of the characteristic orderv _c =O(n ^–x ). This is concluded from the fact that the displacement propagatorD(q, v) assumes the time-dependent Ginzburg-Landau (TDGL) form with a damping coefficient=O(n ^–x ), whenv becomes smaller thanv _c . The exponentx is found to bex=4–d for 2<d<3,x=(d–1)/2 for 3<d<5, andx=2 ford>5. The dynamic critical exponents forv _c (q) and forD(0,v) are derived atT=T _c 0 and toO(1/n). Their values are nontrivial for 2<d<4 and, within the TDGL-region, agree with the those appearing already for frequencies ofO(n ⁰) in TDGL-models with nonconserved order parameter andO(n ⁰)-damping coefficient. The latter case was studied by Halperin, Hohenberg, and Ma in 1972. Even in the TDGL-region, the energy conservation does not affect the dynamic exponents for largen(>2, since the specific heat is finite), but an energy diffusion singularity appears in theQ ²-response function which is related to the basic quantity of the 1/n-method, the effective interactionU _eff. By an estimate of order we find that the damping coefficients resulting from the coupling between the relaxation modes contained inU _eff and the critical modes inD are of ordern ^–w withw>x, such that the coupling between weakly damped critical modes is responsible for the crossover to the TDGL-behavior for largen. The exponentz=d/2, known to be generated by the coupling between order parameter and conservedO(n)-densities in TDGL-models, cannot be seen up to the order calculated. We also point out problems of a microscopic-expansion and comment upon differences between microscopic treatments for displacive transitions and those for the Bose condensation.