Book contents
- Frontmatter
- Contents
- Preface
- Part I Functional integrals and diagram techniques in statistical physics
- Part II Superfluid Bose systems
- Part III Plasma and superfluid Fermi systems
- Part IV Crystals, heavy atoms, model Hamiltonians
- 18 Functional integral approach to the theory of crystals
- 19 Effective interaction of electrons near the Fermi surface
- 20 Crystal structure of a dense electron–ion system
- 21 Quantum crystals
- 22 The theory of heavy atoms
- 23 Functional integral approach to the theory of model Hamiltonians
- References
- Suggested further reading
- Index
18 - Functional integral approach to the theory of crystals
Published online by Cambridge University Press: 01 June 2011
- Frontmatter
- Contents
- Preface
- Part I Functional integrals and diagram techniques in statistical physics
- Part II Superfluid Bose systems
- Part III Plasma and superfluid Fermi systems
- Part IV Crystals, heavy atoms, model Hamiltonians
- 18 Functional integral approach to the theory of crystals
- 19 Effective interaction of electrons near the Fermi surface
- 20 Crystal structure of a dense electron–ion system
- 21 Quantum crystals
- 22 The theory of heavy atoms
- 23 Functional integral approach to the theory of model Hamiltonians
- References
- Suggested further reading
- Index
Summary
In this section we will develop an approach to the microscopic theory of periodic structures in the framework of the functional integration method. This approach was suggested by Kapitonov & Popov (1981) and was developed by Andrianov, Kapitonov & Popov (1982, 1983).
Our starting point will be a system of electrons and ions with the Coulomb interaction. The properties of crystals are determined by the collective excitations (phonons). Clearly, a microscopic theory must describe phonons and their interactions starting from the system of electrons and ions. The functional integral method allows us to realize this aim. The main idea is to go from the initial action of electrons and ions to the effective action functional in terms of the electric potential field φ(x, τ). This field has an immediate physical meaning and provides the collective variable we need.
We can find the static field φ0(x), corresponding to the crystalline structure from the stationary condition for the effective action functional Seff[φ]. If φ0(x) is known we can consider small fluctuations in the vicinity of the stationary point of Seff. In order to do this, we have to expand Seff in this neighbourhood and to separate the quadratic form of φ(x, τ) − φ0(x). It is this quadratic form that defines the spectrum of collective excitations. Forms of the third and higher degrees describe the interaction of these excitations.
The model described below is immediately applicable to the description of metallic hydrogen.
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- Information
- Functional Integrals and Collective Excitations , pp. 147 - 157Publisher: Cambridge University PressPrint publication year: 1988