## MODULATION OF CARR-PURCELL SPIN ECHOES IN MULTI-HALF SPIN SYSTEMS.

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### Date

1967

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Ohio State University

### Abstract

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 $modification^{2}$ of the CPSE experiment, the quantity of interest is the x-component of magnetization, $<I_{x}>$, in the rotating frame. In general, $<I_{x}>$ for the nth echo can be expressed as: $\begin{equation}<I_{x}>_{n} \propto a_{0} + \sum\limits_{1}a\cos(2\pi F_{1} t) + \sum\limits_{m}(-1)^{n} b_{m} \cos(2 \pi F_{m} t),\end{equation}$ with $F_{1\, or\, m} = (2\pi t_{cp})^{-1}|\lambda_{j} - \lambda_{k}|,$ where $t_{cp}$ 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, $a_{1}$'s or $b_{i}$'s can be determined by the use of eigen-functions of $A_{a} \wedge A$ and they are, in general, not equal. $a_{0}$ 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 $t_{cp}$. Analytical expressions for $<I_{x} >_{a}$ 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 ``$A_{n}BX_{x}$'' can be obtained from those for simpler types such as AB, $A_{2}B$ and ABX. Automatic heteronuclear decoupling in CPSE is also discussed for above spin systems. $^{1}$ H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954). $^{2}$ S. Meiboom and D. Gill, Rev. Sci. Instr. 29, 688 (1958). $^{3}$A. Allerhand. J. Chem. Phys. 44, 1 (1966).

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Author Institution: Department of Chemistry, The Ohio State University