This thesis describes electron nuclear double resonance (ENDOR) measurements of hyperfine interactions in various rare-earth systems. A Q-band (8 mm.) ENDOR spectrometer built to perform experiments on Yb171 and Yb173 in CaF2 and Gd155 and Gd157 in ThO2 and Ce0,sub>2. Two other spectrometers are described which were built to perform experiments on Tb3+ in CaWO4, and yttrium ethylsulphate (YES). All the spectrometers operated basically on the principles described by Baker and Williams (1962), using 115 kc/s magnetic field modulation. The nuclear r.f. frequencies were introduced by a single loop for the Yb and Gd experiments, but for the Tb experiments a tuned resonant line was used for a Q-band spectrometer, and a microwave helix for a J-band (2 cm.) spectrometer. The measurements on Yb3+ in the cubic site of CaF2 have provided some very interesting results. Firstly no reliable determination of the crystal field splitting of Yb£+ in cubic CaF2 has been made (Kirton and McLaughlan 1967}. The nuclear moments for Yb171 and Yb173 in diamagnetic Yb2+ ions and in the free atom have been measured by Gossard et. al. (1964) and Olschewski and Otten (1967). A measurement of the effective nuclear g-factor for these isotopes in the paramagnetic Yb3+ ion by ENDOR showed that the effective nuclear g-factor was 64% larger than the true g-factor. This enabled the Γ8 quartet excited state of Yb3+ in CaF2 to be determined to have an energy of 597 ± 6 cm-1 relative to the Γ7 doublet which is the ground state. Using a value for the spin orbit coupling and the value of the one optical transition positively identified by Kirton and McLaughlan to arise from an excited state to the ground state of Yb3+ in cubic sites in CaF2, the crystal field parameters were deduced to be: b4 = 60 B4 = 37.8 cm-1 and b6 = 180 B6 = 9.9 ± 2 cm-1. These values were deduced without accounting for the crystal field admixtures discussed by Bleaney (1964) for isoelectronic Tm2+. The ENDOR measurements on Yb171 (I = ¾) were fitted to a simple spin Hamiltonian, from which A171 and gN171 were found. Yb,sup>173 has I = 5/2, and to fit the spectrum higher order interactions of the type suggested by Ray (1964) had to be included in the spin Hamiltonian. As Yb3+ consists essentially of a single hole in an otherwise complete 4f shell it should be possible to account for these higher order terms by perturbation theory. This was attempted, using combinations of the Zeeman interaction, the magnetic hyperfine interaction and the quadrupole interaction coupling Γ6 and Γ8 levels within the ground manifold to the Γ7 ground doublet. The predicted interactions agreed reasonably well with the observed spin Hamiltonian terms, assuming a quadrupole interaction for the free ion of - 4.7 ± 1.2 mc/s. No such interaction can be observed directly in the ground state of Yb3+ in cubic CaF2, as the effective electronic spin is ½. These higher order perturbations also produce correction terms which appear in the spin Hamiltonian as S.I., thus having the same form as the magnetic hyperfine interaction. It is essential for this work that these terms be correctly evaluated. For Yb171 a correction of - 130 ± 5 kc/s has to be applied by A171 derived from the spin Hamiltonian, whilst for Yb173 the correction is + 121 ± kc/s. The corrected hyperfine interaction values are A171 = 2638.53 ± 0.04 Mc/s, A173 = 727.123 ± 0.06 Mc/s. The ratio of these is A171/A173 - 3.6288 ± 0.0004 leading to a hyperfine structure anomaly of - 0.03 ± 0.03%. From the value for the anomaly deduced by Budick and Snir (1967) for the 6s6p state of the Yb atom of - 0.376 ± 0.020%, a value for the core polarisation in the ion can be deduced. This gives A(s)/A(f) = + 8 ± 8% whilst Bleaney (1963) predicts this ratio to be -1%. The disagreement in sign suggests that further experimental work should be performed, preferably ENDOR at a different EFE frequency. The ENDOR measurements on Gd3+ in ThO2 were a continuation of the work reported by Hurrell (1965). His measurements were made at X-band and the comparison with the results reported here at Q-band showed interesting differences in the A value and in the term A S3.I. To obtain agreement between these experimennts more higher order terms had to be invoked, having the form SIH2 and S3IH2. ENDOR measurements were also performed on Gd157 and Gd155 in CeO2. These crystals appeared to be internally strained and the hyperfine structure EPR lines were unresolved. This resulted in reliable ENDOR measurements being made only on ΔIz = ½ → ½ transitions, as the energies of the Iz = 3/2 levels were smeared out by the crystal field strains. The results for the Gd isotopes gave the following values for the hyperfine interactions : A157 A155 CeO2 15817.2 ± 0.7 Mc/s 12059.8 ± 0.5 Mc/s ThO2 15796.1 ± 0.5 Mc/s No hyperfine structure anomaly was detected. Application of the Sandars and Beck (1965) relativistic theory of hyperfine structure to this system gave reasonable values for the quadrupole interaction. When combined with Bleaney's core polarisation formula the relativistic theory gave values for the hyperfine interactions that were 17% too large. Considerable time was spent searching for the ENDOR of Tb3+ in CaWO4 and YES, using the two spectrometers designed for these experiments. The hyperfine interaction for Tb159 in these crystals is ~6200 Mc/s and thus the design of the nuclear frequency aide of the apparatus required care and is discussed at length. No ENDOR was seen in either host using either apparatus. When using the 2 cm. spectrometer one is likely to encounter ENDOR linewidths of several Mc/s, as in this case the Zeeman interaction is not large compared with the zero field splitting. Tb3+ is a non-Kramers ion and thus has no spin Hamiltonian terms of the form S+I-. This has a serious effect upon the techniques required for the ENDOR experiment. To determine the saturation conditions for ENDOR in CaWO4, a study of the electronic spin-lattice relaxation time as a function of temperature in the range 1-9°K was made using a pulse saturation recovery technique. This was found to be: T1-1 = 2.67 T x 7.7 x 10-4 T7 + 6.65 x 108 exp (-78.7/T) This indicated that the first excited state was at 55 cm-1 which agreed well with a spectroscopic measurement. These results were compared with the published crystal field parameters for this system (Shekun 1965, 1967). The computer programs, experimental data and computed results used tthroughout this thesis are contained in the appendices.