- Spin-Spin Interactions

Spin-Spin Interactions

Previously, we discussed spin-lattice interactions and found that it led to the concept of a longitudinal relaxation time $T_{1}={\tfrac {1}{2W}}$, where $W$ is the average of transition rates out of the upper and lower energy levels. That is, the spin temperature $T_{_{S}}$ will approach the lattice temperature $T$ exponentially with a characteristic time $T_{1}$.

We now concern ourselves with spin-spin interactions.

Since each nucleus possesses a small magnetic dipole moment, there will be a magnetic dipole-dipole interaction between each pair of nuclei. That is, each nuclear magnet finds itself not only in the applied steady magnetic field $H_{_{0}}$ but also in the small local magnetic field $H_{_{loc}}$ produced by its neighboring nuclear magnets. The direction of the local field differs from nucleus to nucleus depending on the spatial configuration of the neighboring nuclei in the lattice and on their magnetic quantum number $m$.

The magnetic field of a magnetic dipole of moment $\mu$ at a distance $r$ is of the order ${\tfrac {\mu }{r^{3}}}$. The field therefore falls off rapidly with increasing $r$, so that only nearest neighbors make important contributions to $H_{_{loc}}$. For a rough estimate of the local field let's take $\mu =\mu _{_{0}}$ and $r=1\mathrm {\AA}$. With this we find that $H_{_{loc}}\approx {\tfrac {\mu _{_{0}}}{r^{3}}}\approx {\text{5 Gauss}}$.

The total magnetic field $H_{_{tot}}=H_{_{0}}+H_{_{loc}}$ will not be the same for each nucleus, but will vary over a range of several Gauss from one nucleus to the next. This implies that the resonance condition will not be perfectly sharp. Instead, the energy levels will be broadened by an amount of order $g\mu _{_{0}}H_{_{loc}}$. If we have a fixed transverse RF field at $\nu _{_{0}}$ the resonance will be found to be spread about $H_{_{0}}$ over a range of values of the order $H_{_{loc}}$. This process is referred to as inhomogeneous broadening.

Since the total magnetic field differs from nucleus to nucleus, there will be a distribution of frequencies of their Larmor precessions ($\omega _{_{0}}=\gamma H_{_{0}}$) covering a range $\delta \omega _{_{0}}\approx \gamma H_{_{loc}}\approx {\tfrac {\mu _{_{0}}^{2}}{\hbar r^{3}}}\approx 10^{4}s^{-1}$.

If two spins have precession frequencies differing by $\delta \omega _{_{0}}$ and are initially in phase, then they will be out of phase in a time $\thicksim {\tfrac {1}{\delta \omega _{_{0}}}}\thicksim 10^{-4}s$.

We have been discussing a physical mechanism by which the nuclear spins interact with each other. Namely, nearest-neighbor-induced local magnetic fields. There is another physical mechanism that can also be important. That is, consider two spins $j$ and $k$. In a steady magnetic field both spins will precess about $H_{_{0}}$ and produce oscillating (at the Larmor frequency) magnetic fields. If spin $j$ produces an oscillating field at spin $k{\text{'s}}$ position it may induce a transition of spin $k$. The energy for this transition comes from spin $j$ so that there is a mutual energy exchange in the process. Since the relative phases of the two spins change in a time of the order of ${\tfrac {1}{\delta \omega _{_{0}}}}$, the correct phasing for this spin-spin exchange process should occur after a time interval of this order and this in turn should determine the lifetime of the spin state. It follows that this spin-spin energy exchange process further broadens the resonance line by an amount of the order of $H_{_{loc}}$.

These two phase disturbing and line broadening processes are only both present when identical nuclei are concerned. For a system of non-identical nuclei, the local field effect is still present but the spin-exchange process is absent (Larmor precession frequencies for different nuclei are quite different).

It is convenient to introduce a spin-spin interaction time $T_{2}$ to describe the lifetime or phase-memory time of a nuclear spin state where $T_{2}\thicksim {\tfrac {1}{\delta \omega _{_{0}}}}\thicksim 10^{-4}$sec. $T_{2}$ is called the transverse relaxation time.

Aside: The reason for naming $T_{1}$ and $T_{2}$ as the longitudinal and transverse relaxation times will become apparent later on.

Note: For liquids and gases, the reorientation and diffusion of the lattice molecules is usually so rapid that the local magnetic field $H_{_{loc}}$ is smoothed out to a very small average value ${\overline {H_{_{loc}}}}$, yielding a quite narrow resonance line.

Although we have discussed here only the basic broadening mechanisms due to spin-spin interactions, we should not overlook other common sources of broadening (listed below).

1) Non-uniformity of the applied magnetic field $H_{_{0}}$ over the assembly of spins.

2) A very short spin-lattice relaxation time $T_{1}$.

3) Electric quadrapole interaction if the spin number exceeds 1/2.

Finally, let's develop a better definition of $T_{2}$. Consider the absorption lineshape as a function of frequency $g(\nu )$. We would want this to be normalized such that

$\int _{0}^{\infty }g(\nu )d\nu =1$. Since the integral is unitless the integrand must be in units of time. The maximum value of the lineshape function $g(\nu )_{max}$ will be large for narrow lines and small for broad lines. The quantity ${\tfrac {1}{g(\nu )_{max}}}$ is then a rough guide to the width of the line. Recall that the linewidth as a function of frequency is of the order $\delta \omega _{_{0}}$. Consequently, the spin-spin interaction time $T_{2}$ is of the order $g(\nu )_{max}$.

We can now make a precise definition of the spin-spin transverse relaxation time $T_{2}={\tfrac {g(\nu )_{max}}{2}}$.

$T_{2}$ is connected only with the peak value of the normalized lineshape curve.