Boltzmann population distribution

Heating a laser-active material populates energy levels only thermally, eventually establishing an excited-state population dictated by the Boltzmann distribution. Population inversions won t be achieved. 

Thennal equilibrium produees a Boltzmann distribution between these energy levels and produees the bulk mielear magnetization of the sample tlirough the exeess population whieh for a sample eontaining a total of N spins is Nytj B lkT. For example for Si in an applied magnetie field of 8.45 T the exeess in the... 

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). 

The intensity distribution among rotational transitions in a vibration-rotation band is governed principally by the Boltzmann distribution of population among the initial states, giving... 

The intensity distribution among the rotational transitions is governed by the population distribution among the rotational levels of the initial electronic or vibronic state of the transition. For absorption, the relative populations at a temperature T are given by the Boltzmann distribution law (Equation 5.15) and intensities show a characteristic rise and fall, along each branch, as J increases. 

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. 

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.

The intensity of a non-degenerate n.m.r. transition between the nuclear Zeeman levels n and m is proportional to the difference in population of levels n and m as given by the Boltzmann distribution. This can be expressed by equation (35). 

At low enough temperatures, only that substitutional isomer which has the lowest ZPVE will be populated. At higher temperatures, however, other isomers may also be found, so that a superposition of several spectra is observed in an ESR experiment. Previously, we have found [1-4] that the abundances of these other isomers are well predicted by a Boltzmann distribution based on the differences in ZPVE. Since the 6-31G(d) and 6-31 lG(d,p) basis sets gave very similar geometries in the present case (see below), the vibrational frequencies and ZPVE for the mono-deuterated isomers were calculated only at the B3-LYP/6-31G(d) level. 

For all known cases of iron-sulfur proteins, J > 0, meaning that the system is antiferromagnetically coupled through the Fe-S-Fe moiety. Equation (4) produces a series of levels, each characterized by a total spin S, with an associated energy, which are populated according to the Boltzmann distribution. Note that for each S level there is in principle an electron relaxation time. For most purposes it is convenient to refer to an effective relaxation time for the whole cluster. 

If the radiofrequency power is too high, the normal relaxation processes will not be able to compete with the sudden excitation (or perturbation), and thermal equilibrium will not be achieved. The population difference (Boltzmann distribution excess) between the energy levels (a and )8) will decrease to zero, and the intensity of the absorption signal will also therefore become zero. 

Such conformational dependence presents challenges and an opportunity. The challenges he in properly accounting for its consequences. In many cases, exact conformational energetics and populations in a sample may be unknown, and the nature of the sample inlet may sometimes also mean that a Boltzmann distribution cannot be assumed. Introducing this uncertainty into the data modeling process produces some corresponding uncertainty in the theoretical interpretation of data... 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

Fig. 1.1 Schematic representation of the population difference of spins at different magnetic field strengths. The two different spin quantum number values of the ]H spin, +34 and -34, are indicated by arrows. Spins assume the lower energy state preferentially, the ratio bet-ween upper and lower energy level being given by the Boltzmann distribution.

The ratio of populations at equilibrium is given by the Boltzmann distribution ... 

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