The energy levels of coupled systems
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص524-525
2025-12-10
37
The energy levels of coupled systems
It will be useful for later discussions to consider an NMR spectrum in terms of the energy levels of the nuclei and the transitions between them. In NMR, letters far apart in the alphabet (typically A and X) are used to indicate nuclei with very different chemical shifts; letters close together (such as A and B) are used for nuclei with similar chemical shifts. We shall consider first an AX system, a molecule that contains two spin-
nuclei A and X with very different chemical shifts in the sense that the difference in chemical shift corresponds to a frequency that is large compared to their spin–spin coupling. The energy level diagram for a single spin- nucleus and its single transition were shown in Fig. 15.3, and nothing more needs to be said. For a spin- AX system there are four spin states:
αAαX αAβX βAαX βAβX .
The energy depends on the orientation of the spins in the external magnetic field, and if spin–spin coupling is neglected
E=−γh(1−σA) BmA−γh(1−σX) BmX=−hνAmA−hνXmX
Fig. 15.11The energy levels of an AX system. The four levels on the left are those of the two spins in the absence of spin–spin coupling. The four levels on the right show how a positive spin–spin coupling constant affects the energies. The transitions shown are for β←αof A or X, the other nucleus (X or A, respectively) remaining unchanged. We have exaggerated the effect for clarity; in practice, the splitting caused by spin–spin coupling is much smaller than that caused by the applied field.
Where νA and νX are the Larmor frequencies of A and X and mA and mX are their quantum numbers. This expression gives the four lines on the left of Fig. 15.11. The spin–spin coupling depends on the relative orientation of the two nuclear spins, so it is proportional to the product mAmX. Therefore, the energy including spin–spin coupling is
E=−hνAmA−hνXmX+hJmAmX
If J>0, a lower energy is obtained when mAmX0) is shown on the right of Fig. 15.11. We see that the ααandββstates are both raised by1–4hJand that theαβandβαstates are both lowered by1–4hJ. When a transition of nucleus A occurs, nucleus X remains unchanged. Therefore, the A resonance is a transition for which ∆mA=+1 and ∆mX=0. There are two such transitions, one in which βA←αA occurs when the X nucleus is αX, and the other in whichβA←αA occurs when the X nucleus is βX. They are shown in Fig. 15.11 and in a slightly different form in Fig. 15.12. The energies of the transitions are
∆E=hνA± hJ
Therefore, the A resonance consists of a doublet of separation J centred on the chemical shift of A (Fig. 15.13). Similar remarks apply to the X resonance, which consists of two transitions according to whether the A nucleus is αorβ (as shown in Fig. 15.12). The transition energies are
∆E=hνX± hJ
It follows that the X resonance also consists of two lines of separation J, but they are centred on the chemical shift of X (as shown in Fig. 15.13).
Fig. 15.12An alternative depiction of the energy levels and transitions shown in Fig. 15.11. Once again, we have exaggerated the effect of spin–spin coupling.
Fig. 15.13The effect of spin–spin coupling on an AX spectrum. Each resonance is split into two lines separated by J. The pairs of resonances are centred on the chemical shifts of the protons in the absence of spin–spin coupling.
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