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Mathematical Research in Materials Science Opportunities and Perspectives (1993) / Chapter Skim
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7 PROCESSING, FABRICATION, AND EVALUATION
Pages 77-87

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From page 77...
... This chapter presents an assortment (that is far from comprehensive) of materials processing, fabrication, and evaluation topics that involve applications of or offer research opportunities in the mathematical sciences.
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From page 78...
... During subsequent processing, further diffusion takes place. Many of the steps described above have been modeled to various degrees of accuracy by partial differential equations that describe the relevant physics.
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From page 79...
... The mathematical model is an extension of the equations for a silicon semiconductor device; it includes diffusion equations for the trapped holes and electrons, which depend on the density-of-states and the occupation functions (Straw et al., 1991~. There are various experimental power laws suggested by simulations (Straw and Hack, 1988)
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From page 80...
... Outstanding mathematical sciences research issues in these contexts are summarized in Chapter 3. OTHER PROCESSING There are many other materials processing settings where opportunities for mathematical sciences research exist.
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From page 81...
... For ultrasonics, this description requires computation of the transducer radiation pattern, refraction of the beam at surfaces, the beam profile, and the propagation characteristics in the host material, including effects of material anisotropy, attenuation, and diffraction losses. Detailed modeling of field-flaw interactions, which generate the response function of the measurement system, must also be included as well as information on conditions that produce statistical noise and add uncertainty to the results.
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From page 82...
... Layered materials, such as materials with a ceramic coating to protect a metallic substrate or with directly bonded homogeneous layers, have major disadvantages here: high thermal and residual stresses and poor bond strength (Houck, i988~. However, the recently developed technique of intentionally grading the composition of an interracial region (Kerrihara et al., 1990)
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From page 83...
... Some examples of NLO effects are frequency conversions (such as higher harmonics, generation of sum and difference frequencies, optical parametric oscillations, and stimulated Raman scattering) , nonlinear refractive indexes and related self-action effects (such as self-focusing, self-trapping, self-bending, optical bistability, solitons in fibers)
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From page 84...
... Bulk semiconductors are some of the most universal nonlinear optical materials; under proper conditions they can demonstrate almost any possible nonlinear optical effect and can be used for applications from 0.3- to 12-,um wavelengths with a variety of materials systems. The nonlinear phenomenon in bulk semiconductors that has attracted most of the attention in the last decade is the nonlinear refractive index and related effects.
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From page 85...
... Films can grow into fascinating self-similar and fractal structures. Kinetic growth theories based, for example, on random-deposition models and ballistic-deposition models are very attractive from both the physical and the mathematical point of view in that they exhibit dynamic scaling properties governed by simple functions with constant characteristic exponents.
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From page 86...
... Photorefractive materials are electro-optic crystals in which absorption of photons triggers a charge migration resulting in a spatial modulation of the material's refractive index through a space-charge-field-induced electro-optic effect. There are some exciting mathematical problems related to the theory of propagation of conjugated waves in photorefractive materials, especially in self-pumped conjugators.
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From page 87...
... A number of multiparameter theoretical models of the single-atom response to intensive optical fields in HHG have been suggested; most of them reproduce qualitatively the gross picture of the process. The most successful theoretical approach thus far has been direct numerical simulation using Hartry-Slater approximation of the atomic Schrodinger equations for many-electron atoms.
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Key Terms

  • mathematical sciences
  • nonlinear optical
  • mathematical challenge
  • crystal growth
  • nonlinear refractive

  • photorefractive materials
  • materials processing
  • mathematical analysis
  • differential equations
  • refractive index

  • partial differential
  • mathematical modeling
  • thermomechanical properties
  • nonlinear interaction
  • optical bistability

  • layered materials
  • mathematical challenges
  • major features
  • optical materials
  • frequency conversion

  • response function
  • integrodifferential equations
  • mathematical research
  • refractive indexes
  • variable coefficients

  • laser sources
  • nlo effects
  • nondestructive evaluation
  • mathematical model
  • gradient materials


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