*Robert E. Newnham*

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780198520757
- eISBN:
- 9780191916601
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198520757.003.0017
- Subject:
- Earth Sciences and Geography, Geochemistry

The physical properties discussed thus far are linear relationships between two measured quantities. This is only an approximation to the truth, and often not a very good approximation, especially ...
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The physical properties discussed thus far are linear relationships between two measured quantities. This is only an approximation to the truth, and often not a very good approximation, especially for materials near a phase transformation. A more accurate description can be obtained by introducing higher order coefficients. To illustrate nonlinearity we discuss electrostriction, magnetostriction, and higher order elastic, and dielectric effects. These phenomena are described in terms of fourth and sixth rank tensors. Many of the recent innovations in the field of electroceramics have exploited the nonlinearities of material properties with factors such as electric field, mechanical stress, temperature, or frequency. The nonlinear dielectric behavior of ferroelectric ceramics (Fig. 15.1), for example, has opened up new markets in electronics and communications. In these materials the electric polarization saturates under high fields. Electric displacement Di varies with applied electric field Ej as … Di = εijEj + εijkEjEk + εijklEjEkEl +· · ·, … where εij is the dielectric permittivity and εijk and εijkl are higher order terms. The data in Fig. 15.1 were collected for a relaxor ferroelectric in its paraelectric state above Tc where the symmetry is centrosymmetric. Therefore the third rank tensor εijk is zero, and the shape of the curve is largely controlled by the first and third terms. For cubic crystals, the fourth rank tensor εijkl is similar in form to the elastic constants discussed in Chapter 13. Tunable microwave devices utilize nonlinear dielectrics in which the polarization saturates as in Fig. 15.1. By applying a DC bias the dielectric constant can be adjusted over a wide range.
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The physical properties discussed thus far are linear relationships between two measured quantities. This is only an approximation to the truth, and often not a very good approximation, especially for materials near a phase transformation. A more accurate description can be obtained by introducing higher order coefficients. To illustrate nonlinearity we discuss electrostriction, magnetostriction, and higher order elastic, and dielectric effects. These phenomena are described in terms of fourth and sixth rank tensors. Many of the recent innovations in the field of electroceramics have exploited the nonlinearities of material properties with factors such as electric field, mechanical stress, temperature, or frequency. The nonlinear dielectric behavior of ferroelectric ceramics (Fig. 15.1), for example, has opened up new markets in electronics and communications. In these materials the electric polarization saturates under high fields. Electric displacement Di varies with applied electric field Ej as … Di = εijEj + εijkEjEk + εijklEjEkEl +· · ·, … where εij is the dielectric permittivity and εijk and εijkl are higher order terms. The data in Fig. 15.1 were collected for a relaxor ferroelectric in its paraelectric state above Tc where the symmetry is centrosymmetric. Therefore the third rank tensor εijk is zero, and the shape of the curve is largely controlled by the first and third terms. For cubic crystals, the fourth rank tensor εijkl is similar in form to the elastic constants discussed in Chapter 13. Tunable microwave devices utilize nonlinear dielectrics in which the polarization saturates as in Fig. 15.1. By applying a DC bias the dielectric constant can be adjusted over a wide range.

*Antony Bryant*

- Published in print:
- 2017
- Published Online:
- January 2017
- ISBN:
- 9780199922604
- eISBN:
- 9780190652548
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199922604.003.0012
- Subject:
- Psychology, Social Psychology

Sampling in GTM – purposive, convenience, snow-ball samples at the start, then theoretical sampling at later stages. The differnecies between substantive and formal grounded theories. Kearney’s ...
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Sampling in GTM – purposive, convenience, snow-ball samples at the start, then theoretical sampling at later stages. The differnecies between substantive and formal grounded theories. Kearney’s discussion of the two types, and the differnet ways in which they can be derived. Status Passage as the only jointly published formal grounded theory by Glaser and Strauss. The meaning of Theoretical Saturation and some misconceptions. Theoretical coding and taking the outcome of a grounded theory back to the literature. A student’s understanding of Theoretical Saturation. Theoretical Saturation illustrated in the context of Crème Brûlée. Differentiating Substantive grounded theories from formal grounded theories.Less

Sampling in GTM – purposive, convenience, snow-ball samples at the start, then theoretical sampling at later stages. The differnecies between substantive and formal grounded theories. Kearney’s discussion of the two types, and the differnet ways in which they can be derived. *Status Passage* as the only jointly published formal grounded theory by Glaser and Strauss. The meaning of *Theoretical Saturation* and some misconceptions. Theoretical coding and taking the outcome of a grounded theory back to the literature. A student’s understanding of Theoretical Saturation. Theoretical Saturation illustrated in the context of *Crème Brûlée*. Differentiating Substantive grounded theories from formal grounded theories.

*Guang S. He*

- Published in print:
- 2014
- Published Online:
- December 2014
- ISBN:
- 9780198702764
- eISBN:
- 9780191772368
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198702764.003.0009
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter describes the principles of major nonlinear and ultrahigh resolution laser spectroscopic techniques, including saturation spectroscopy, two-photon spectroscopy, coherent Raman ...
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This chapter describes the principles of major nonlinear and ultrahigh resolution laser spectroscopic techniques, including saturation spectroscopy, two-photon spectroscopy, coherent Raman spectroscopy, nonlinear polarization spectroscopy, and laser cooling and trapping spectroscopy. All these spectral techniques are based on the use of one or several laser beams (at least one of which is tunable), and there is no need for any ordinary spectrometers or dispersion elements (such as prisms, gratings, or Fabry–Perot etalon). The most remarkable advantages of these novel spectroscopic techniques are their Doppler-free capability and ultrahigh spectral resolution, which enable researchers to measure hyperfine structures, isotope shifts, Stark and Zeeman splitting, and to establish new optical frequency standards (atomic clocks) as well.Less

This chapter describes the principles of major nonlinear and ultrahigh resolution laser spectroscopic techniques, including saturation spectroscopy, two-photon spectroscopy, coherent Raman spectroscopy, nonlinear polarization spectroscopy, and laser cooling and trapping spectroscopy. All these spectral techniques are based on the use of one or several laser beams (at least one of which is tunable), and there is no need for any ordinary spectrometers or dispersion elements (such as prisms, gratings, or Fabry–Perot etalon). The most remarkable advantages of these novel spectroscopic techniques are their Doppler-free capability and ultrahigh spectral resolution, which enable researchers to measure hyperfine structures, isotope shifts, Stark and Zeeman splitting, and to establish new optical frequency standards (atomic clocks) as well.

*Craig M. Bethke*

- Published in print:
- 1996
- Published Online:
- November 2020
- ISBN:
- 9780195094756
- eISBN:
- 9780197560778
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195094756.003.0007
- Subject:
- Earth Sciences and Geography, Geochemistry

Aqueous geochemists work daily with equations that describe the equilibrium points of chemical reactions among dissolved species, minerals, and gases. To study an individual reaction, a geochemist ...
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Aqueous geochemists work daily with equations that describe the equilibrium points of chemical reactions among dissolved species, minerals, and gases. To study an individual reaction, a geochemist writes the familiar expression, known as the mass action equation, relating species’ activities to the reaction’s equilibrium constant. In this chapter we carry this type of analysis a step farther by developing expressions that describe the conditions under which not just one but all of the possible reactions in a geochemical system are at equilibrium. We consider a geochemical system comprising at least an aqueous solution in which the species of many elements are dissolved. We generally have some information about the fluid’s bulk composition, perhaps directly because we have analyzed it in the laboratory. The system may include one or more minerals, up to the limit imposed by the phase rule (see Section 3.4), that coexist with and are in equilibrium with the aqueous fluid. The fluid's composition might also be buffered by equilibrium with a gas reservoir (perhaps the atmosphere) that contains one or more gases. The gas buffer is large enough that its composition remains essentially unchanged if gas exsolves from or dissolves into the fluid. How can we express the equilibrium state of such a system? A direct approach would be to write each reaction that could occur among the system’s species, minerals, and gases. To solve for the equilibrium state, we would determine a set of concentrations that simultaneously satisfy the mass action equation corresponding to each possible reaction. The concentrations would also have to add up, together with the mole numbers of any minerals in the system, to give the system’s bulk composition. In other words, the concentrations would also need to satisfy a set of mass balance equations. Such an approach, however, is unnecessarily difficult to carry out. Dissolving even a few elements in water produces many tens of species that need be considered, and complex solutions contain many hundreds of species. Each species represents an independent variable, namely its concentration, in our scheme. For any but the simplest of chemical systems, the problem would contain too many unknown values to be solved conveniently.
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Aqueous geochemists work daily with equations that describe the equilibrium points of chemical reactions among dissolved species, minerals, and gases. To study an individual reaction, a geochemist writes the familiar expression, known as the mass action equation, relating species’ activities to the reaction’s equilibrium constant. In this chapter we carry this type of analysis a step farther by developing expressions that describe the conditions under which not just one but all of the possible reactions in a geochemical system are at equilibrium. We consider a geochemical system comprising at least an aqueous solution in which the species of many elements are dissolved. We generally have some information about the fluid’s bulk composition, perhaps directly because we have analyzed it in the laboratory. The system may include one or more minerals, up to the limit imposed by the phase rule (see Section 3.4), that coexist with and are in equilibrium with the aqueous fluid. The fluid's composition might also be buffered by equilibrium with a gas reservoir (perhaps the atmosphere) that contains one or more gases. The gas buffer is large enough that its composition remains essentially unchanged if gas exsolves from or dissolves into the fluid. How can we express the equilibrium state of such a system? A direct approach would be to write each reaction that could occur among the system’s species, minerals, and gases. To solve for the equilibrium state, we would determine a set of concentrations that simultaneously satisfy the mass action equation corresponding to each possible reaction. The concentrations would also have to add up, together with the mole numbers of any minerals in the system, to give the system’s bulk composition. In other words, the concentrations would also need to satisfy a set of mass balance equations. Such an approach, however, is unnecessarily difficult to carry out. Dissolving even a few elements in water produces many tens of species that need be considered, and complex solutions contain many hundreds of species. Each species represents an independent variable, namely its concentration, in our scheme. For any but the simplest of chemical systems, the problem would contain too many unknown values to be solved conveniently.