Population difference

Note that is the population difference between the upper and lower states having all the population in the lower state corresponds to = -1 while having a completely inverted population (i.e. no population in the lower state) corresponds to f3 = +1. 

Application of an oscillating magnetic field at the resonance frequency induces transitions in both directions between the two levels of the spin system. The rate of the induced transitions depends on the MW power which is proportional to the square of oi = (the amplitude of the oscillating magnetic field) (see equation (bl.15.7)) and also depends on the number of spins in each level. Since the probabilities of upward ( P) a)) and downward ( a) p)) transitions are equal, resonance absorption can only be detected when there is a population difference between the two spin levels. This is the case at thennal equilibrium where there is a slight excess of spins in the energetically lower p)-state. The relative population of the two-level system in thennal equilibrium is given by the Boltzmaim distribution... 

Figure C3.2.14. Electron population difference x(t) = - P (0 for tliree electron transfer reactions in the...

Application. A change was made in a manufacturing procedure for component parts. Samples were taken during the last week of operations with the old procedure and during the first week of operations with the new procedure. Determine whether the proportional numbers of defects for the respective populations differ on the basis of the sample information. 

From the discussion in Section 1.1, can you summarize the factors affecting the population difference between the lower energy state Nf) and the upper energy state Nf) and how is the population difference related to the NMR signal strength ... 

As mentioned in the text, there is only a slight excess of nuclei in the ground state (about 13 in a million protons at 100 MHz). Would you expect in the case of a C-NMR experiment for the same population difference to prevail ... 

The population difference between two energy levels a and /3 is directly proportional to the energy difference AE,... 

No. Since the magnetogyric ratio of C is roughly one-fourth that of H, the population difference between the two states (a and )3) of C nuclei will therefore be about 64 times [(1/4) = 1/64] less than that of the H nuclei. 

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. 

Suppose, for clarity, we add a common factor [(V2)jn + (V2)yc] to all four energy levels so the energy difference between them does not change. The values of these energy levels will then become y, yc, Jh, and yH + yc-Since yn is about four times yc, let us also assume that yn = 4 and yc = 1. The populations of the four energy levels will then be 0, 1, 4, and 5. The population difference between the two C spin states before the application of the H polarization transfer pulse corresponds to the lower energy state minus the upper energy state, i.e. 1—0=1 or 5 — 4=1. 

The pulse sequence used in the reverse DEPT experiment is shown in Fig. 2.16. Presaturation of the protons removes all H magnetization and pumps up the C population difference due to nOe. Broad-band decoupling of the C nuclei may be carried out. The final spectrum obtained is a one-dimensional H-NMR plot that contains only the H signals to which polarization has been transferred—for instance, from the enriched C nucleus. 

When the pulse is applied to nucleus S, the population levels connected by S transitions are rapidly equalized, but there is no immediate change in the difference in population levels connected by the I transitions. Figure 4.2 shows that on application of the pulse, levels 2 and 4 acquire the same populations as levels 1 and 3, respectively, but the population difference... 

If we now examine the result of this population adjustment on the I transitions we find that the upper population of each I transition has decreased by x while the lower level of each I transition has increased by x the population difference between the upper and lower states (levels 1 and 3) connected by the /i transition is... 

Similarly, the population difference between the states connected by the transition (levels 2 and 4) is... 

Hence the population difference between the lower and upper energy states of the two I transitions becomes d + x, as compared to the original difference of d at equilibrium. Thus an intensification of the lines for nucleus 1 will be observed by an amount corresponding to this increased difference x. This is the positive nuclear Overhauser effect that is encountered in small, rapidly tumbling molecules, in which Wj is the predominant relaxation pathway. 

In large molecules that tumble slowly, the predominant relaxation pathway is via This is shown schematically in Fig. 4.2c. A part of the population x is now transferred from the /3q state to the a)3 state. This causes an increase in the population of the upper level of one 1 transition (/], level 3) and a decrease in the lower population level of the other I transition (4, level 2). As a result, the population difference between the lower and upper levels of each I transition is reduced to d — x (i.e., level 1 — level 3, or level 2 — level 4, becomes d — x). The reduction in population difference by x as compared to the equilibrium situation (Fig. 4.2a)... 

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. 

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