Equilibrium populations

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

A4) Bond angle bending makes a nonnegligible contribution to conformational entropy and can affect computed equilibrium populations [11]. 

The emission signal corresponding to benzene confirms that it is formed by a free-radical process. As in steady-state EPR experiments, the enhanced emission and absorption are observed only as long as the reaction is proceeding. When the reaction is complete or is stopped in some way, the signals rapidly return to their normal intensity, because equilibrium population of the two spin states is rapidly reached. 

If only single-quantum transitions (h, I2, S], and S ) were active as relaxation pathways, saturating S would not affect the intensity of I in other words, there will be no nOe at I due to S. This is fairly easy to understand with reference to Fig. 4.2. After saturation of S, the fMjpula-tion difference between levels 1 and 3 and that between levels 2 and 4 will be the same as at thermal equilibrium. At this point or relaxation processes act as the predominant relaxation pathways to restore somewhat the equilibrium population difference between levels 2 and 3 and between levels 1 and 4 leading to a negative or positive nOe respectively. 

The equilibrium population is said to have a structure when a substantial fraction of the molecules adopts similar conformations. But the phrase lacking structure does not imply that individual molecules comprising the ensemble lack a conformation rather, the population is too heterogeneous to be readily characterized using a coherent, structure-based descriptor. The unfolded state resists ready characterization because it is so diverse. Typical biophysical methods report ensemble-... 

It is possible to create a population of Schottky defects that is much higher than the equilibrium population that is based on Eq. (7.32). If a crystal is heated to high temperature, lattice vibrations become more pronounced, and eventually ions begin to migrate from their lattice sites. If the crystal is quickly cooled, the extent of the motion of ions decreases rapidly so that ions that have moved from their lattice sites cannot return. As a result, the crystal will contain a population of Schottky defects that is much higher than the equilibrium population at the lower temperature. If a crystal of KC1 is prepared so that it contains some CaCl2 as an impurity, incorporating a Ca2+ ion in the crystal at a K+ site... 

Here, the concept of linkage implies only that each intermolecular noncovalent bond is sufficiently large compared with kTto withstand ambient thermal collisions. Thus, for near-standard-state conditions (where kT 0.6kcal mol-1), even weak noncovalent interactions of 1-2 kcal mol-1 may be adequate to yield supramolecular complexes with stable equilibrium populations, thereby becoming true constituent units of the phase of lowest free energy. 

The examples cited above are only two of the many possible cases of H-bond isomerization. Because of the low kinetic barriers separating these species, equilibration of H-bonded isomer populations to limiting thermodynamic values is generally expected to be much faster than for covalent isomers. Methods of quantum statistical thermodynamics can be used to calculate partition functions and equilibrium population distributions for H-bonded isomers,41 just as in the parallel case for covalent isomers and conformers. 

On enthalpic grounds alone, under ambient conditions the Boltzmann factor for the chain pentamer would be reduced by about 1010 relative to that for the cyclic pentamer, showing that the former topology contributes negligibly to the equilibrium population distribution. 

EQUILIBRIUM POPULATION OF SELF-INTERSTITIALS IN A MONATOMIC CRYSTAL... 

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