*E. L. Wolf*

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198769804
- eISBN:
- 9780191822636
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198769804.003.0004
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

Protons in the Sun’s core are a dense plasma allowing fusion events where two protons initially join to produce a deuteron. Eventually this leads to alpha particles, the mass-four nucleus of helium, ...
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Protons in the Sun’s core are a dense plasma allowing fusion events where two protons initially join to produce a deuteron. Eventually this leads to alpha particles, the mass-four nucleus of helium, releasing kinetic energy. Schrodinger’s equation allows particles to penetrate classically forbidden Coulomb barriers with small but important probabilities. The approximation known as Wentzel–Kramers–Brillouin (WKB) is used by Gamow to predict the rate of proton–proton fusion in the Sun, shown to be in agreement with measurements. A simplified formula is given for the power density due to fusion in the plasma constituting the Sun’s core. The properties of atomic nuclei are briefly summarized.Less

Protons in the Sun’s core are a dense plasma allowing fusion events where two protons initially join to produce a deuteron. Eventually this leads to alpha particles, the mass-four nucleus of helium, releasing kinetic energy. Schrodinger’s equation allows particles to penetrate classically forbidden Coulomb barriers with small but important probabilities. The approximation known as Wentzel–Kramers–Brillouin (WKB) is used by Gamow to predict the rate of proton–proton fusion in the Sun, shown to be in agreement with measurements. A simplified formula is given for the power density due to fusion in the plasma constituting the Sun’s core. The properties of atomic nuclei are briefly summarized.

*Wolfgang Schmickler*

- Published in print:
- 1996
- Published Online:
- November 2020
- ISBN:
- 9780195089325
- eISBN:
- 9780197560563
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195089325.003.0026
- Subject:
- Chemistry, Physical Chemistry

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, ...
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The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations are not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H2O)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunctions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. Let us denote by R the coordinates of all the nuclei involved, those of the central ion, its ligarids, and the surrounding solvation sphere, and by r the coordinates of all electrons.
Less

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations are not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H2O)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunctions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. Let us denote by R the coordinates of all the nuclei involved, those of the central ion, its ligarids, and the surrounding solvation sphere, and by r the coordinates of all electrons.