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# and receptivity

First of all let us see which nuclei may be used for an NMR measurement. The first consideration is that in general there is more than one isotope for each atom and the fundamental prequisite is that the chosen isotope does possess a non zero spin, since this quantity ranges in practice from 0 to 9/2.

 The nucleus used in the first pioneering experiments was that of hydrogen ($^1H$, the proton), which has spin $I=\frac 1 2$, its abundance is 99.9885 %, its magnetogyric ratio is $^1\gamma = 26.75\, 10^7\,\mbox{rad/(Ts)} = 42.577 \,\mbox{MHz/T}$. It provides a very intense signal from any hydrogen containing material and it is the standard reference for signal intensity, or receptivity, as we shall define below. The relevant properties of all other nuclei may be found on the web NMR periodic table. The table on the left shows the $I=\frac 1 2$ isotopes.

So, how is receptivity defined? When we apply a large static field $\mbox{\it \bf B}$ to a non magnetic sample its nuclear magnetic moments respond to the field independently, to a very good approximation, behaving like the ideal paramagnet. Hence the equilibrium nuclear magnetization is given by the Curie law:

$(1) \qquad\qquad \mbox{\it \bf M} = na\frac {\gamma^2\hbar^2}{3k_BT} \mbox{\it \bf B}$

where $n$ is the density of atoms and $a$ is the natural abundance of the selected isotope. The electromotive force across the pick-up coil will be proportional to the derivative of M(t), hence:

$(2) \qquad\qquad \varepsilon \propto \omega M = na\frac {\hbar^2\gamma^3B^2}{3k_BT}$

 This tells us that, with equal volumes of sample, inside the same coil (tuning of the resonant circuit allowing), the NMR signal amplitude scales with abundance of the isotope, density, the square of the resonance frequency, the third power of the magnetogyric ratio and the inverse of temperature. Of course this is true only if all the nuclei (i.e. their entire spectrum) are being irradiated and detected, a condition which is not always met. The height of the bars is proportional to the receptivity of the main isotopes