Knowledge Bank

Density matrix procedures for the analysis of Carr-Purcell spin echoes $(CPSE)1$ in the presence of nuclear spin-spin coupling have been developed for several multi-half-spin systems. Since we assume the use of Mciboom-Gill $modification2$ of the CPSE experiment, the quantity of interest is the x-component of magnetization, , in the rotating frame. In general, for the nth echo can be expressed as: with $F1orm=(2\pi tcp)-1|\lambda j-\lambda k|,$ where $tcp$ is the separation between the $180\circ$ pulses and $\lambda i$ is the ith eigenvalue in the $A\wedge A$, $representation.3$ In Eq. (1) the amplitude factors, $a1$'s or $bi$'s can be determined by the use of eigen-functions of $Aa\wedge A$ and they are, in general, not equal. $a0$ is the unmodulated contribution. The modulation frequencies and their relative amplitudes are functions not only of the relative chemical shifts and coupling constants, but also strongly dependent on $tcp$. Analytical expressions for have been obtained in multi-half-spin systems of the type $A_{a}BX_{x}$'' where a and x are arbitrary numbers of half-spin nuclei. It is discussed that the general expression of $<I_{x}>_{n}$ for the type $AnBXx$'' can be obtained from those for simpler types such as AB, $A2B$ and ABX. Automatic heteronuclear decoupling in CPSE is also discussed for above spin systems. $\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}$ H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954). $\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ S. Meiboom and D. Gill, Rev. Sci. Instr. 29, 688 (1958). $\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}$A. Allerhand. J. Chem. Phys. 44, 1 (1966).