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Lattice Vibrations Part III Solid State Physics 355

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Back to Dispersion Curves We know we can measure the phonon dispersion curves - We know we can measure the phonon dispersion curves - the dependence of the phonon frequencies upon the wavevector q. To calculate the heat capacity, we begin by summing over all the energies of all the possible phonon modes, multiplied by the Planck Distribution. Planck Distribution sum over all wavevectors sum over all polarizations

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Density of States number of modes unit frequency D( )

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Density of States: One Dimension determined by the dispersion relation If the ends are fixed, what modes, or wavelengths, are allowed?

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Density of States: One Dimension # of wavelengths wavelengthwavevector 0.5 2L2L2L2L /L /L 1L 2 /L 1.5 2L/3 3 /L 2 L/2 4 /L

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Density of States: One Dimension To calculate the density of states, use number of modes unit frequency D( ) There is one mode per interval q = / L with allowed values... So, the number of modes per unit range of q is L /.

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Density of States: One Dimension To generalize this, go back to the definition...the number of modes is the product of the density of states and the frequency unit. There is one mode for each mobile atom.

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Density of States: One Dimension monatomic lattice diatomic lattice Knowing the dispersion curve we can calculate the group velocity, d /dq. Near the zone boundaries, the group velocity goes to zero and the density of states goes to infinity. This is called a singularity.

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Periodic Boundary Conditions No fixed atoms – just require that u(na) = u(na + L). This is the periodic condition. The solution for the displacements is The allowed q values are then,

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Density of States: 3 Dimensions Lets say we have a cube with sides of length L. Apply the periodic boundary condition for N 3 primitive cells:

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Density of States: 3 Dimensions There is one allowed value of q per volume (2 /L 3 ) in q space or allowed values of q per unit volume of q space, for each polarization, and for each branch. The total number of modes for each polarization with wavevector less than q is qzqz qyqy qxqx

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Debye Model for Heat Capacity Debye Approximation: For small values of q, there is a linear relationship =vq, where v is the speed of sound....true for lowest energies, long wavelengths This will allow us to calculate the density of states.

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Debye Model for Heat Capacity

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qDqD

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Debye Temperature is related to 1. The stiffness of the bonds between atoms 2. The velocity of sound in a material, v 3. The density of the material, because we can write the Debye Temperature as:

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Debye Model for Heat Capacity How did Debye do??

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Debye Model for Heat Capacity

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Einstein's oscillator treatment of specific heat gave qualitative agreement with experiment and gave the correct high temperature limit (the Law of Dulong and Petit). The quantitative fit to experiment was improved by Debye's recognition that there was a maximum number of modes of vibration in a solid. He pictured the vibrations as standing wave modes in the crystal, similar to the electromagnetic modes in a cavity which successfully explained blackbody radiation. Debye Model for Heat Capacity

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ω D represents the maximum frequency of a normal mode in this model. ω D is the energy level spacing of the oscillator of maximum frequency (or the maximum energy of a phonon). It is to be expected that the quantum nature of the system will continue to be evident as long as The temperature in gives a rough demarcation between quantum mechanical regime and the classical regime for the lattice.

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Typical Debye frequency: (a) Typical speed of sound in a solid ~ 5×10 3 m/s. A simple cubic lattice, with side a = 0.3 nm, gives ω D 5×10 13 rad/s. (b) We could assume that k max /a, and use the linear approximation to get ω D v sound k max 5×10 13 rad/s. A typical Debye temperature: θ D 450 K Most elemental solids have θ D somewhat below this.

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Measuring Specific Heat Capacity Differential scanning calorimetry (DSC) is a relatively fast and reliable method for measuring the enthalpy and heat capacity for a wide range of materials. The temperature differential between an empty pan and the pan containing the sample is monitored while the furnace follows a fixed rate of temperature increase/decrease. The sample results are then compared with a known material undergoing the same temperature program.

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Measuring Specific Heat Capacity

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