Boltzmann energy levels

Details of the FT process are given elsewhere [1], but briefly, an NMR spectrometer generates pulsed RF radiation (typically of microsecond duration) which excites nuclei in the sample to non-Boltzmann energy levels the resulting decay to the ground state generates RF radiation... 

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... 

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the... 

The Boltzmann distribution gives the number of particles n, in each energy level e, as ... 

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

The Boltzmann formula relates the entropy of a substance to the number of arrangements of molecules that result in the same energy when many energy levels are accessible, this number and the corresponding entropy are large. 

When the length of the box is increased at constant temperature (with T > 0), more energy levels become accessible to the molecules because the levels now lie closer together (Fig. 7.9b). The disorder has increased and we are less sure about which energy level any given molecule occupies. Therefore, the value of W increases as the box is lengthened and, by the Boltzmann formula, the entropy increases, too. The same argument applies to a three-dimensional box as the volume of the box increases, the number of accessible states increases, too. 

We can also use the Boltzmann formula to interpret the increase in entropy of a substance as its temperature is raised (Eq. 2 and Table 7.2). We use the same par-ticle-in-a-box model of a gas, but this reasoning also applies to liquids and solids, even though their energy levels are much more complicated. At low temperatures, the molecules of a gas can occupy only a few of the energy levels so W is small and the entropy is low. As the temperature is raised, the molecules have access to larger numbers of energy levels (Fig. 7.10) so W rises and the entropy increases, too. 

The effect of temperature upon the situation in Fig. 5-5 is to modify the Boltzmann distribution. Lowering the temperature depopulates the higher-lying energy levels in favour of the lower. Therefore, susceptibility increases with decreasing temperature. Quantitative studies of the simple (first-order )... 

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. 

Figure 4.2 Energy levels and populations for an IS system in which nuclei I and S are not directly coupled with each other. This forms the basis of the nuclear Overhauser enhancement effect. Nucleus S is subjected to irradiation, and nucleus I is observed, (a) Population at thermal equilibrium (Boltzmann population).

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

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.

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

Fig. 5.2 Radial distribution curves, Pv

Consider a simple two energy level system, e.g., an S - 1/2 system. For a sample of realistic size at thermal equilibrium n0 molecules are in the lowest energy state and / in the highest state according to the Boltzmann distribution... 

In the quantum mechanical formulation of electron transfer (Atkins, 1984 Closs et al, 1986) as a radiationless transition, the rate of ET is described as the product of the electronic coupling term J2 and the Frank-Condon factor FC, which is weighted with the Boltzmann population of the vibrational energy levels. But Marcus and Sutin (1985) have pointed out that, in the high-temperature limit, this treatment yields the semiclassical expression (9). 

We have discussed the transition moment (the quantum mechanical control of the strength of a transition or the rate of transition) and the selection rules but there is a further factor to consider. The transition between two levels up or down requires either the lower or the upper level to be populated. If there are no atoms or molecules present in the two states then the transition cannot occur. The population of energy levels within atoms or molecules is controlled by the Boltzmann Law when in local thermal equilibrium ... 

Local thermal equilibrium (LTE) is an assumption that allows for the molecules to be in equilibrium with at least a limited region of space and remains an assumption when using the Boltzmann law for the relative populations of energy levels. The LTE assumption notwithstanding, observation of a series of transitions in the spectrum and measurement of their relative intensities allows the local temperature to be determined. We shall see an example of this in Section 4.4 where the Balmer temperature of a star is derived from the populations of different levels in the Balmer series. 

The rotational spectrum of a molecule involves transitions between energy levels, say the R(8) transition / = 8 and J = 9, but if there are no molecules rotating in the J = 8 level then there can be no R(8) transition. The local thermal collisions will populate some of the higher J levels in a general principle called equipartition. The general expression is the Boltzmann Law, given by ... 

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