Nuclear Magnetic Resonance Saturation and Rotary Saturation in Solids

Nuclear spin-lattice relaxation times of ${\mathrm{Al}}^{27}$ in pure Al and ${\mathrm{Cu}}^{63}$ in annealed pure Cu have been measured with a nuclear induction spectrometer, by the method of saturation. The experimental values of ${T}_{1}$ are 4.1\ifmmode\pm\else\textpm\fi{}0.8 milliseconds for ${\mathrm{Al}}^{27}$ and 3.0\ifmmode\pm\else\textpm\fi{}0.6 milliseconds for ${\mathrm{Cu}}^{63}$, in reasonable agreement with theory.The dispersion mode of the nuclear resonance was also observed, and it was found that ${\ensuremath{\chi}}^{\ensuremath{'}}$ (the real part of the rf susceptibility) does not saturate at the same level as the absorption, ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$, but remains roughly constant out to a radio-frequency field intensity of about 2 gauss. Both ${\ensuremath{\chi}}^{\ensuremath{'}}$ and ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$ become narrower and nearly Lorentzian in shape above saturation. When the dc magnetic field modulation is increased from 14 to 41 cps the phase of the dispersion signal lags behind the modulation, presumably because the modulation period is then comparable to ${T}_{1}$. Large dispersion signals above saturation have also been observed for the ${\mathrm{Na}}^{23}$ resonance in NaCl.This behavior of the dispersion mode is in conflict with the predictions of Bloembergen, Purcell, and Pound and of the Bloch equations. The validity of these theories is re-examined, and it is concluded that although they are applicable to nuclear resonance in liquids and gases, and to solids at small rf intensities, they contain incorrect assumptions as applied to solids at high rf power levels. The theory of Bloembergen, Purcell, and Pound is based on an assumption equivalent to that of a spin temperature. It is shown that the spin state cannot be strictly described by a spin temperature because the phases of the spin quantum states are not incoherent, as required by the temperature concept. The transverse decay of the nuclear magnetization predicted by the Bloch equations is shown to be partially forbidden by energy and entropy considerations if a large rf field at the resonance frequency is continuously applied to the solid.A theory is developed which is applicable only to solids at rf magnetic field intensities well above the saturation level and which is in reasonable agreement with the experimental observations. The Hamiltonian is transformed to a coordinate system rotating at the frequency of the rf field. The resulting time-dependent parts of the spin-spin interaction are nonsecular perturbations on the time-independent part, and can therefore be ignored. Statistical mechanics is applied to the remaining stationary spin Hamiltonian; specifically it is assumed that the spin system is in its most probable macrostate (a canonical distribution of quantum states) with respect to the transformed spin Hamiltonian. This assumption is justified because the transformed spin Hamiltonian is effectively time independent and the spin-lattice interaction is small, and it is analogous to assumptions basic to classical acoustics and fluid mechanics. The spin-lattice interaction merely determines the expectation value of the transformed spin Hamiltonian, which can be readily calculated under the assumption that the expectation value of the spin angular momentum of each spin is relaxed independently to its thermal equilibrium value by the lattice in time ${T}_{1}$. Both fast and slow modulation of the dc magnetic field can be treated."Rotary saturation" is observed by applying an audio-frequency magnetic field to the sample in the dc field direction while observing the dispersion derivative at resonance with a large rf field ${H}_{1}$. When the audio-frequency approaches $\ensuremath{\gamma}{H}_{1}$ the dispersion signal decreases and goes through a minimum. The effect is easily treated theoretically in solids, liquids and gases by using a rotating coordinate system, and is a rotary analogue of ordinary saturation. It is a convenient method for calibrating rf magnetic fields and appears potentially capable of providing useful information on the solid state. Experimental data on rotary saturation are presented and discussed.