Book contents
- Frontmatter
- Contents
- Preface
- List of symbols
- 1 Morphology of a crystal surface
- 2 Surface free energy, step free energy, and chemical potential
- 3 The equilibrium crystal shape
- 4 Growth and dissolution crystal shapes: Frank's model
- 5 Crystal growth: the abc
- 6 Growth and evaporation of a stepped surface
- 7 Diffusion
- 8 Thermal smoothing of a surface
- 9 Silicon and other semiconducting materials
- 10 Growth instabilities of a planar front
- 11 Nucleation and the adatom diffusion length
- 12 Growth roughness at long lengthscales in the linear approximation
- 13 The Kardar-Parisi-Zhang equation
- 14 Growth without evaporation
- 15 Elastic interactions between defects on a crystal surface
- 16 General equations of an elastic solid
- 17 Technology, crystal growth and surface science
- Appendix A From the discrete Gaussian model to the two-dimensional Coulomb gas
- Appendix B The renormalization group applied to the two-dimensional Coulomb gas
- Appendix C Entropic interaction between steps or other linear defects
- Appendix D Wulff's theorem finally proved
- Appendix E Proof of Frank's theorem
- Appendix F Step flow with a Schwoebel effect
- Appendix G Dispersion relations for the fluctuations of a train of steps
- Appendix H Adatom diffusion length ℓs and nucleation
- Appendix I The Edwards-Wilkinson model
- Appendix J Calculation of the coefficients of (13.1) for a stepped surface
- Appendix K Molecular beam epitaxy, the KPZ model, the Edwards-Wilkinson model, and similar models
- Appendix L Renormalization of the KPZ model
- Appendix M Elasticity in a discrete lattice
- Appendix N Linear response of a semi-infinite elastic, homogeneous medium
- Appendix O Elastic dipoles in the z direction
- Appendix P Elastic constants of a cubic crystal
- References
- Index
16 - General equations of an elastic solid
Published online by Cambridge University Press: 07 May 2010
- Frontmatter
- Contents
- Preface
- List of symbols
- 1 Morphology of a crystal surface
- 2 Surface free energy, step free energy, and chemical potential
- 3 The equilibrium crystal shape
- 4 Growth and dissolution crystal shapes: Frank's model
- 5 Crystal growth: the abc
- 6 Growth and evaporation of a stepped surface
- 7 Diffusion
- 8 Thermal smoothing of a surface
- 9 Silicon and other semiconducting materials
- 10 Growth instabilities of a planar front
- 11 Nucleation and the adatom diffusion length
- 12 Growth roughness at long lengthscales in the linear approximation
- 13 The Kardar-Parisi-Zhang equation
- 14 Growth without evaporation
- 15 Elastic interactions between defects on a crystal surface
- 16 General equations of an elastic solid
- 17 Technology, crystal growth and surface science
- Appendix A From the discrete Gaussian model to the two-dimensional Coulomb gas
- Appendix B The renormalization group applied to the two-dimensional Coulomb gas
- Appendix C Entropic interaction between steps or other linear defects
- Appendix D Wulff's theorem finally proved
- Appendix E Proof of Frank's theorem
- Appendix F Step flow with a Schwoebel effect
- Appendix G Dispersion relations for the fluctuations of a train of steps
- Appendix H Adatom diffusion length ℓs and nucleation
- Appendix I The Edwards-Wilkinson model
- Appendix J Calculation of the coefficients of (13.1) for a stepped surface
- Appendix K Molecular beam epitaxy, the KPZ model, the Edwards-Wilkinson model, and similar models
- Appendix L Renormalization of the KPZ model
- Appendix M Elasticity in a discrete lattice
- Appendix N Linear response of a semi-infinite elastic, homogeneous medium
- Appendix O Elastic dipoles in the z direction
- Appendix P Elastic constants of a cubic crystal
- References
- Index
Summary
Do not imagine you can abdicate:
Before you reach the frontier you are caught;
Others have tried it and will try again
To finish what they did not begin.
W. H. AudenAs long as possible we have postponed a general study of elasticity and its partial differential equations. Here they are!
The elastic equilibrium of a solid is generally treated in the continuum approximation. The strain satisfies certain equations in the bulk, and other equations at the surface. This set of equations has an infinite number of solutions, and the correct one is that which minimizes a given thermodynamic potential or free energy. This minimization is not needed for a semi-infinite solid because the good solution in this case is the one which vanishes at infinity.
The power of continuous elasticity theory is limited. In particular it is not appropriate to investigate the surface relaxation, i.e. the change in the atomic distance near the surface. Nevertheless, the continuum approximation allows for spectacular predictions, for instance the Asaro-Tiller-Grinfeld instability, which is one of the major obstacles to layer-by-layer heteroepitaxial growth.
Memento of elasticity in a bulk solid
In this section, the theory of linear elasticity in a homogeneous solid away from the surface will be recalled.
In order to write the condition for mechanical equilibrium, one has to consider the forces acting on a volume δV of the solid (Fig. 16.1). There may be an external force δfext, and there is a force produced by the part of the solid outside δV.
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- Physics of Crystal Growth , pp. 249 - 276Publisher: Cambridge University PressPrint publication year: 1998