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Lattice Vibrations Part IV. Solid State Physics 355. Thermal Expansion. Anharmonic effects can be important for physical properties. As you heat up the solid, internal energy of the lattice increases as k B T; and the lattice expands. Thermal Expansion.

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## Lattice Vibrations Part IV

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**Lattice VibrationsPart IV**Solid State Physics 355**Thermal Expansion**• Anharmonic effects can be important for physical properties. • As you heat up the solid, internal energy of the lattice increases as kBT; and the lattice expands.**Thermal Expansion**The average displacement is determined from... After some manipulation... If A is zero, there is no thermal expansion.**Thermal Expansion**Dilatometer**Thermal Expansion**alumina (Al2O3)**Negative Thermal Expansion**Zirconium tungstate exhibits “negative thermal expansion” from 0.3 K up to at least 1400 K. The structure of ZrW2O8 consists of a framework of ZrO6 octahedra and WO4 tetrahedra linked at corners, but with one of the corners of the WO4 tetrahedra remaining unlinked.**Negative Thermal Expansion**Many tetrahedrally bonded materials show negative thermal expansion at low temperatures; for example, the thermal expansion of ice Ih becomes negative below 80 K. The dynamics of ice, even in its natural hexagonal form, are still a puzzle despite many decades of work. The combination of the rotational disorder and the complexity of the inter-molecular forces make modeling the system difficult. H2O, Si, Ge, ZnSe, GaP, GaAs**Thermal Conductivity**What is heat? Heat is the spontaneous flow of energy from an object at a higher temperature to an object at a lower temperature.**TH**TC average particle velocity mean free path specific heat capacity per unit volume Thermal Conductivity • Thermal conduction is a diffusion process and proceeds via the random movement of electrons and phonons. • These particles carry energy from one part of the solid, where the internal energy is higher toward a region where the internal energy is lower. • From the kinetic theory of gases...**Thermal Conductivity**• As a phonon moves a distance d, it will reduce the temperature by T as it carries energy away. • This change in temperature is • The amount of energy carried by each phonon is then, • The number of phonons passing through a unit area per unit time is the**Thermal Conductivity**• The net flux of energy is then,**Laser Flash Diffusivity**______ ___ __ _________**Phonon Scattering**• Phonon scattering with other phonons is the result of anharmonic effects. • If the forces between atoms were purely harmonic, there would be no mechanism for collisions between different phonons; and the mean free path would be limited solely by geometrical influences such as boundaries and imperfections.**Phonon Scattering: U Processes**The physical result of all this is that a phonon comes along and “experiences” a different local “stiffness” due to the strain caused by another phonon.

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