Boltzmann distribution excited states

Standardizing the Method Equation 10.34 shows that emission intensity is proportional to the population of the excited state, N, from which the emission line originates. If the emission source is in thermal equilibrium, then the excited state population is proportional to the total population of analyte atoms, N, through the Boltzmann distribution (equation 10.35). 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

Relaxation refers to all processes which regenerate the Boltzmann distribution of nuclear spins on their precession states and the resulting equilibrium magnetisation along the static magnetic field. Relaxation also destroys the transverse magnetisation arising from phase coherenee of nuelear spins built up upon NMR excitation. 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

If the conditions for Forster transfer are not applicable, then the theory must be extended. There is recently experimental evidence that coherent energy transfer participates in photosynthesis [74, 75], In this case, the participating molecules are very close together. The excited state of the donor does not completely relax to the Boltzmann distribution before the energy can be shared with the acceptor, and the transfer can no longer be described by a Forster mechanism. We will not discuss this case. There has been active discussion of coherent transfer and very strong interactions in the literature for a longer time [69], and references can be found in some more recent papers [70-72, 76, 77],... 

Just as above, we can derive expressions for any fluorescence lifetime for any number of pathways. In this chapter we limit our discussion to cases where the excited molecules have relaxed to their lowest excited-state vibrational level by internal conversion (ic) before pursuing any other de-excitation pathway (see the Perrin-Jablonski diagram in Fig. 1.4). This means we do not consider coherent effects whereby the molecule decays, or transfers energy, from a higher excited state, or from a non-Boltzmann distribution of vibrational levels, before coming to steady-state equilibrium in its ground electronic state (see Section 1.2.2). Internal conversion only takes a few picoseconds, or less [82-84, 106]. In the case of incoherent decay, the method of excitation does not play a role in the decay by any of the pathways from the excited state the excitation scheme is only peculiar to the method we choose to measure the fluorescence (Sections 1.7-1.11). 

The quantum Boltzmann distribution only applies between allowed energy levels of the same family and each type of energy has its own characteristic partition function, that can be established by statistical methods and describes the response of a system to thermal excitation. If the total number of particles N, is known, the Boltzmann distribution may be used to calculate the number, or fraction, of molecules in each of the allowed quantum states. For any state i... 

The low-energy, low-frequency range for NMR transitions corresponds to a small change in energy, AE. This has implications for the population of excited states, the Boltzmann distribution. For a spin-1/2 nuclei with AE= (iBq// and I = 1/2, equation 3.27 applies. Because N+ N, one can write equation 3.28 ... 

Relaxation is an inherent property of all nuclear spins. There are two predominant types of relaxation processes in NMR of liquids. These relaxation processes are denoted by the longitudinal (Ti) and transverse (T2) relaxation time constants. When a sample is excited from its thermal equihbrium with an RF pulse, its tendency is to relax back to its Boltzmann distribution. The amount of time to re-equilibrate is typically on the order of seconds to minutes. T, and T2 relaxation processes operate simultaneously. The recovery of magnetization to the equilibrium state along the z-axis is longitudinal or the 7 relaxation time. The loss of coherence of the ensemble of excited spins (uniform distribution) in the x-, y-plane following the completion of a pulse is transverse or T2... 

At thermal equilibrium, according to the Boltzmann distribution (see Section 2.3), the population of atoms and molecules in the excited state can never exceed the population in the ground state for a simple two-level system. 

Electrons thermally excited from the valence band (VB) occupy successively the levels in the conduction band (CB) in accordance with the Fermi distribution function. Since the concentration of thermally excited electrons (10 to 10 cm" ) is much smaller than the state density of electrons (10 cm ) in the conduction band, the Fermi function may be approximated by the Boltzmann distribution function. The concentration of electrons in the conduction band is, then, given by the following integral [Blakemore, 1985 Sato, 1993] ... 

Little is known about the fluorescence of the chla spectral forms. It was recently suggested, on the basis of gaussian curve analysis combined with band calculations, that each of the spectral forms of PSII antenna has a separate emission, with Stokes shifts between 2nm and 3nm [133]. These values are much smaller than those for chla in non-polar solvents (6-8 nm). This is due to the narrow band widths of the spectral forms, as the shift is determined by the absorption band width for thermally relaxed excited states [157]. The fluorescence rate constants are expected to be rather similar for the different forms as their gaussian band widths are similar [71], It is thought that the fluorescence yields are also probably rather similar as the emission of the sj tral forms is closely approximated by a Boltzmann distribution at room temperature for both LHCII and total PSII antenna [71, 133]. 

However, simultaneously there is emission of infrared radiation from the Boltzmann distribution of molecules in excited states, which leads to a negative energy component. This emission process depends not only on the concentration of the gas but very sensitively on the temperature since this determines the population of the excited states that emit (Eq. (A)). As we shall see in some specific cases below, it is the balance between these two at any given altitude that determines the changes in fluxes and the ultimate impact of a change in a greenhouse gas concentration. 

When a system is in thermodynamic equilibrium the level population, i.e. the number of atoms A in the excited state, is given by the Boltzmann distribution law ... 

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