Lecture 27. Debye Model of Solids, Phonon Gas

1. ħ 1 3 2 3N In 1907, Einstein developed the first quantum-mechanical model of solids that was able to qualitatively describe the low-T heat capacity of the crystal lattice. Although this was a crucial step in the right direction, the model was too crude. Lecture 27. Debye Model of Solids, Phonon Gas In Einstein’s model, all oscillators are identical, and their frequencies are the same: However, Einstein’s model ignores the fact that the atomic vibrations are coupled together: the potential energy of an atom in the crystal depends on the distance from its neighbors: This energy is not a sum of single-particle energies. Thus, the calculation of the partition function may look rather difficult. But a system of Ncoupled three-dimensional oscillators is equivalent to a system of 3Nindependent one-dimensional oscillators. the price to be paid is that the independent oscillators are not of the same frequency; the normal modes of vibration of a solid have a wide range of frequencies. These modes are not related to the motion of single atoms, but to the collective motion of all atoms in the crystal – vibrational modes or sound waves.

2. Einstein’s Model of a Solid In 1907 Einstein, in the first application of quantum theory to a problem other than radiation, modeled a solid body containing N atoms as a collection of 3N harmonic oscillators. The partition function of a single oscillator: The oscillators are independent of each other, thus: The mean energy: This looks familiar: the same energy would have a photon of frequency . The internal energy is not a directly measurable quantity, and instead we measure the heat capacity: equipartition high T (kBT>>h): Limits: low T (kBT<<h): The Einstein model predicts much too low a heat capacity at low temperatures!

3. Debye’s model (1912) starts from the opposite point of view, treating the solid as a continuum, i.e., the atomic structure is ignored. A continuum has vibrational modes of arbitrary low frequencies, and at sufficiently low T only these low frequency modes are excited. These low frequency normal modes are simply standing sound waves. Debye’s Theory of the Heat Capacity of Solids Nobel 1936 If we quantize this elastic distortion field, similar to the quantization of the e.-m. field, we arrive at the concept of phonons, the quanta of this elastic field. For the thermal phonons, the wavelength increases with decreasing T : cS – the sound velocity These low-energy modes remain active at low temperatures when the high-frequency modes are already frozen out. Large values of  that correspond to these modes justify the use of a continuum model. There is a close analogy between photons and phonons: both are “unconserved” bosons. Distinctions: (a) the speed of propagation of phonons (~ the speed of sound waves) is by a factor of 105 less than that for light, (b) sound waves can be longitudinal as well as transversal, thus 3 polarizations (2 for photons), and (c) because of discreteness of matter, there is an upper limit on the wavelength of phonons – the interatomic distance.

4. For a macroscopic crystal, the spectrum of sound waves is almost continuous, and we can treat  is a continuous variable. As in the case of photons, we start with the density of states per unit frequency g(). The number of modes per unit volume with the wave number < k: Density of States in Debye Model - multiplied by 3 since a sound wave in a solid can have three polarizations (two transverse and one longitudinal). - this eq. only holds for sufficiently low  (= large wavelengths and the continuous approximation is valid). There is also an upper cut-off for the frequencies (  interatomic distances), the so-called Debye frequencyD, which depends on the density n: cS ~3 km/s, a~0.2 nm, D~1013 Hz Each normal mode is a quantized harmonic oscillator. The mean energy of each mode: and is the total energy per unit volume. The U0 term comes from the zero-point motion of atoms. It reduces the cohesive energy of the solid (the zero point motion in helium is sufficient to prevent solidification at any T at normal pressure), but since it does not depend on T, it does not contribute to C. Note that we ignored this term for phonons, where it is . In QED, this unobservable term is swept under the rug by the process known as renormalization.

5. The Heat Capacity in Debye’s Model At low temperatures, we can choose the upper limit as  (the high-frequency modes are not excited, the energy is too low). How low should be T : The low-T heat capacity: Thus, Debye’s model predicts that in the limit of sufficiently low T, the heat capacity due to vibrations of the crystal lattice (in a metal electrons also contribute to C) must vary as T3, and not as 2exp(- h/kBT), as in Einstein’s model. (Roughly speaking, the number of phonons ~ T3, their average energy is proportional to T). At high temperatures, all the modes are excited (the number of phonons does not increase any more), and the heat capacity approaches the equipartition limit, C=3NkB.

6. Debye Temperature The material-specific parameter is the sound speed. If the temperature is properly normalized, the data for different materials collapse onto a universal dependence: The normalization factor is called the Debye temperature: It is related to the maximum frequency D, the Debye frequency: The higher the sound speed and the density of ions, the higher the Debye temperature. However, the real phonon spectra are very complicated, and D is better to treat as an experimental fitting parameter.

7. Problem (blackbody radiation) The spectrum of Sun, plotted as a function of energy, peaks at a photon energy of 1.4 eV. The spectrum for Sirius A, plotted as a function of energy, peaks at a photon energy of 2.4 eV. The luminosity of Sirius A (the total power emitted by its surface) is by a factor of 24 greater than the luminosity of the Sun. How does the radius of Sirius A compare to the Sun’s radius? The temperature, according to Wien’s law, is proportional to the energy that corresponds to the peak of the photon distribution.

8. Final 2006 (blackbody radiation) The frequency peak in the spectral density of radiation for a certain distant star is at 1.7 x 1014 Hz. The star is at a distance of 1.9 x 1017 m away from the Earth and the energy flux of its radiation as measured on Earth is 3.5x10-5 W/m2. a) (5) What is the surface temperature of the star? b) (5) What is the total power emitted by 1 m2 of the surface of the star? c) (5) What is the total electromagnetic power emitted by the star? d) (5) What is the radius of the star? (a) (b) (c) (d)

9. Problem 2006 (blackbody radiation) • The cosmic microwave background radiation (CMBR) has a temperature of approximately 2.7 K. • (a) (5) What wavelength λmax(in m) corresponds to the maximum spectral density u(λ,T) of the cosmic background radiation? • (5) What frequency max(in Hz) corresponds to the maximum spectral density u(,T) of the cosmic background radiation? • (5) Do the maxima u(λ,T) and u(,T) correspond to the same photon energy? If not, why? • (d) (15) What is approximately the number of CMBR photons hitting the earth per second per square meter [i.e. photons/(s·m2)]? (a) (b) (c) The maxima u(λ,T) and u(,T) do not correspond to the same photon energy. As  increases, the frequency range included in unit wavelength interval increases as 2, moving the peak to shorter wavelengths:

10. Problem 2006 (blackbody radiation) cont. (d)

11. Problem (blackbody radiation) The planet Jupiter is a distance of 7.78 x 1011 meters from the Sun and is of radius 7.15 x 107 meters. Assume that Jupiter is a blackbody even though this is not entirely correct. Recall that the solar power output of Sun is 4 x 1026 W. a) What is the energy flux of the Sun's radiation at the distance of Jupiter's orbit? b) Recognizing that Sun's energy falls on the geometric disk presented to the Sun by the planet, what is the total power incident on Jupiter from the Sun? c) Jupiter rotates at a rather rapid rate (one revolution per 0.4 earth days) and therefore all portions of the planet absorb energy from the Sun. Hence all portions of the surface of this planet radiate energy outward. On the basis of this information find the surface temperature of Jupiter.

12. d) Estimates of the surface temperature of Jupiter indicate that it is clearly above 500 Kelvins. On the basis of your answer to c) what might you conclude about this planet? There must be some other source of energy to produce a surface temperature of 500 K.

13. Problem (blackbody radiation) Tungsten has an emissivity of 0.3 at high temperatures. Tungsten filaments operate at a temperature of 4800 K. a) At what frequency does a tungsten filament radiate the most energy? b) What is the power/unit surface emitted by the tungsten filament? c) If the surface area of a tungsten filament is 0.01 cm2 what is the power output of the bulb? d) What is the energy flux of radiation emitted by the filament 2 meters from the bulb? e) What fraction of radiation incident on a tungsten filament is reflected? [Answers: a) 2.88 x 1014 Hz, b) 9.03 x106 W/m2, c) 9.03 watts, d) 0.180 watts, and e) 70%.]

14. Problem (radiation pressure) • At what temperature will the pressure of the photon gas be equal to 105 Pa (= one bar )? • At what temperature will the pressure of the photon gas be equal to 10-5 Pa? • The temperature at the Sun’s center is 107 K. What is the pressure of the radiation? Compare it to the pressure of the gas at the center of the Sun, which is 1011 bar. • Calculate the pressure of the Sun’s radiation on the Earth’s surface, given that the power of the radiation reaching earth from the Sun is 1400 W/m2. (a) P=105 Pa (b) P=10-5 Pa (c) - at this T, the pressure of radiation is still negligible in the balance for mechanical equilibrium (d)

15. Problem (blackbody radiation) The black body temperature is 3000K. Find the power emitted by this black body within the wavelength interval  = 1 nm near the maximum of the spectrum of blackbody radiation. the power emitted by a unit area in all directions: near the maximum: