Multiplets for Spin I = 1/2 Nuclides Coupled to Quadrupolar Nuclides

When a spin I = 1/2 nuclide is coupled to a qudrupolar nuclide of spin S, one will observe one of three things: (1) a sharp singlet; (2) a sharp multiplet consisting of (2NS + 1) lines, where N is the number of equivalent quadrupolar nuclides or; (3) a broadened signal intermediate between a sharp singlet and a sharp multiplet.  An example of a sharp singlet is the proton decoupled ¹³C spectrum of CHCl₃.  In this case, the lifetimes for each of the (2S + 1) Zeeman states of the quadrupolar ³⁵Cl and ³⁷Cl nuclides are much shorter than the reciprocal of the J coupling interaction with the ¹³C.  As a result, the  ¹³C "sees" an average state for the ³⁵Cl and ³⁷Cl nuclides and appears as a sharp singlet (neglecting any isotope effects).  This phenomenon is quite common and is sometimes referred to as "self-decoupling".  An example of the a sharp multiplet is the ¹³C spectrum of CDCl₃. In this case, the lifetimes of each of the three Zeeman states of the deuterium are much longer than the reciprocal of the J coupling interaction with the ¹³C.  As a result, the ¹³C "sees" each of the three Zeeman states of the deuterium and appears as a 1:1:1 triplet.  Examples of broadened signals, intermediate between sharp singlets and sharp multiplets are often observed in the signals of ¹³C bound to ¹⁴N ( I = 1 ).  In these cases, the lifetimes of the three Zeeman states of the ¹⁴N are of the same order as the reciprocal of the J coupling interaction with the ¹³C.  It is very interesting to note that the lifetimes for each of the quadrupolar Zeeman states are not necessarily the same.  As a result, for cases where the lifetimes are comparable to the reciprocal of the J coupling interaction, one can observe differential broadening in the lines of the multiplet.  This can be seen for the  ¹³C signals for tetraalkyl ammonium salts  or the ¹H signal for dilute HDO where, in each case, the outer two lines of the observed triplets are broadened more than the central line such that the intensity (height) of the lines does not appear to be 1:1:1.  A more dramatic example is shown in the figure below for the ¹³C spectrum of K₃[Co(CN)₆].

 For this compound, the lifetimes of the ¹⁴N Zeeman states are sufficiently short such that the ¹⁴N is self-decoupled from the ¹³C.  The lifetimes of the ⁵⁹Co ( I = 7/2, 100% naturally abundant) Zeeman states, on the other hand, are comparable (although somewhat longer than) the reciprocal of the J coupling interaction with the ¹³C.  As a result, an eight line multiplet is observed with equal integrals for each of the component lines.  In this case, the lifetimes of the eight ⁵⁹Co Zeeman states are not equal and one can see that lines 2 and 7 of the multplet are the most broadened.  Lines  3 and 6 are the second most broadened followed by lines 4 and 5.  Lines 1 and 8 are the least broadened and therefore the highest in intensity.  The calculation of the relevant lineshapes has been presented in the early (and no so early) literature.^1-4

1. J.A. Pople, Mol. Phys. 1, 168 (1958).
2. J. Bacon, R.J. Gillespie and J.W. Quail. Can. J. Chem. 41, 3063 (1963).
3. M. Suzuki and R. Kubo, Mol. Phys. 7, 201 (1964).
4. N.C. Pyper, Mol. Phys. 19, 161 (1970).
5. P. Kofod, J. Magn. Res. A. 119, 212 (1996).