Harmonic oscillator at thermal equilibrium

Harmonic oscillators are often used as approximate models for realistic systems. A common application is their use as convenient models for the thermal environments of systems of interest (see Section 6.5). Such models are mathematically simple, yet able to account for the important physical attributes of a thennal bath temperature, coupling distribution over the bath normal modes, and characteristic timescales. Their prominence in such applications is one reason why we study them in such detail in this chapter. 

The treatment of quantum systems in thermal equilibrium, and of systems interacting with their thennal environments is expanded on in Chapter 10. For now 

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A collection of N harmonic oscillators at thermal equilibrium at absolute temperature T is shown by statistical mechanics to have the thermodynamic energy... 

To remove fast oscillations we work in the interaction picture introducing new operators Aj = fi/(z)exp(—ikjz)- The damping of vibration modes is modelled as coupling of each vibration mode to the broad reservoir of harmonic oscillators in thermal equilibrium [140]. The two important parameters are damping constants jv and mean number of chaotic phonons (nVj),j= 1,2. Finally we arrive at [127] the following ... 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

The idea of a thermodynamic temperature scale was first proposed in 1854 by the Scottish physicist William Thomson, Lord Kelvin [iv]. He realized that temperature could be defined independently of the physical properties of any specific substance. Thus, for a substance at thermal equilibrium (which can always be modeled as a system of harmonic oscillators) the thermodynamic temperature could be defined as the average energy per harmonic oscillator divided by the Boltzmann constant. Today, the unit of thermodynamic temperature is called kelvin (K), and is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. 

Thermal unimolecular reactions usually exhibit first-order kinetics at high pressures. As pointed out originally by Lindemann [1], such behaviour is found because collisionally energised molecules require a finite time for decomposition at high pressures, collisional excitation and de-excitation are sufficiently rapid to maintain an equilibrium distribution of excited molecules. Rice and Ramsperger [2] and, independently, Kassel [3] (RRK), realised that a detailed theory must take account of the variation of decomposition rate of an excited molecule with its degree of internal excitation. Kassel s theory is still widely used and is valid for the chosen model of a set of coupled, classical, harmonic oscillators. 

Next, consider the relaxation of a diatomic molecule, essentially a single oscillator of frequency mq interacting with its thermal environment, and contrast its behavior with a polyatomic molecule placed under similar conditions. The results of the simple hannonic relaxation model of Section 9.4 may give an indication about the expected difference. The harmonic oscillator was shown to relax to the thermal equilibrium defined by its environment, Eq. (9.65), at a rate given by (cf. Eq. (9.57))... 

Figure 5.3 Thermal distributions for clusters of 4, 7, 9, 12 and 16 atoms in equilibrium with a heat bath at 150 K, assuming a model of coupled harmonic oscillators. The real distributions of sodium clusters can be expected to be sharper than shown here, as anharmonicities and melting have been neglected...

ZPE and thermal and entropic corrections at the appropriate experimental temperatures can be calculated using the frequencies in conjimction with the standard textbook formulas for the statistical thermodynamics of an ideal gas under the harmonic oscillator/rigid rotor approximation. Equations (4) and (5) relates the rate constant and equilibrium constant with the Gibbs free energy, which can be described in terms of the enthalpy (H) and the entropy (S) in the following equation ... 

The model of harmonic oscillations of the Mossbauer atom as described above, is quite satisfactory for solids at low temperatures. However, at higher temperatures the smaU amplitude motion of the atom around its equilibrium site may become unstable such that the atom has a flnite probability of making a sudden jump to an adjacent lattice site. Although this type of motion can in principle occur even at very low temperatures via the mechanism of quantum tunnelling, there is little evidence for this phenomenon with Mossbauer isotopes. The diffusive motion can therefore be considered in terms of thermally activated jumps... 

The force disappears only for y = 0. This, however, does not imply that in a thermal equilibrium the end-groups are coupled together. A harmonic oscillator with a stiffness constant b which is in contact with a heat bath at a temperature T shows a non-vanishing mean-squared displacement ( / ). Straightforward application of Boltzmann-statistics yields... 

Figure C.l shows schematically the potential energy of bonded atoms. The continuous line is for a more rigid material since it shows a deeper potential. Parameter ro corresponds to the equilibrium distance at T= 0 K. This temperature cannot be produced, so for clarity we take this temperature as a reference point. At r> OK, atoms oscillate around their equilibrium position. It is important to note that the potential is not symmetric. For small oscillations, one generally neglects this phenomenon and takes a parabolic function in the form Ep = (ll2)k(r-ro) also called the harmonic potential. Force is then F=-k(r-ro). Under such an assumption, no thermal expansion is found (Figure C.2).

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