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The magnetic field dependence in the amplitude of de Haas-van Alphen or magnetoresistance oscillations introduces magnetic field dependent phase shifts which are not included in the standard formation of the Lifshitz-Kosevich theory. It is found that the magnetic field dependence of the oscillatory amplitude introduces an additional signal which is pi /2 out of phase in 1/H with respect to the signal which would be observed if the amplitude of the oscillation is strictly independent of magnetic field strength. A number of examples are presented which illustrate the magnitude of this magnetic field dependent amplitude effect when the large amplitude modulation field technique is employed to measure the frequency of the quantum oscillations.

The results of an experiment at 4.2 K using the magnetoresistance oscillations caused by the interferometer trajectories in magnesium are reported. The magnesium crystal used in this experiment is believed to have an impurity concentration of ∼1 ppm. The quantum state lifetime of electron states traversing the interferometer was determined to be .01±.002 nsec. In addition, the values of the three magnetic breakdown parameters needed to describe the size of the spin orbit energy gaps occurring near the AHL zone plane in magnesium were found to be:H1=2.90±.1KG,H2=12.5±.5KG,H3=4.25±.25KG. The presence of higher harmonics was also observed in the magnetoresistance oscillations and the higher harmonics were used to allow an experimentally meaningful value ofH3 to be determined. It is presently believed that the sensitivity of the interferometer to quantum state lifetimes can be extended by using the higher harmonics and hence information about electron-phonon and possibly electron-electron interactions will be obtained.

The “on-axis” and “off-axis” regimes of quantum transport discussed theoretically in the preceding paper in this journal have been observed experimentally. The observation of the “on-axis” regime required magnetic field alignment to within an accuracy of ±0.002° of the basal plane. The data for the “off-axis” regime indicate that the electron quantum state lifetime for this crystal is about 5×10−10 sec at 4.2 K. A comparison between the predictions of first principles theory and the experimental results in these two regimes shows that the theory not only predicts the qualitative behavior correctly but also provides a quantitatively accurate description for the transverse magnetoresistance.

Studies of the de Haas---van Alphen spectra arising from the two-dimensional lattice of magnetic-breakdown---coupled orbits in magnesium contain spectral features which disagree qualitatively with the accepted theory of Falicov and Stachowiak. Their theory, which was thought to be equivalent to Pippard's theory for the magnetic band structure, is shown to be inequivalent.

The effect of changes in the lattice parameters on the Fermi surface of zinc is calculated. The results of the calculations are in reasonably good agreement with experimental data.

The magnetoresistance of extremely pure, strain-free magnesium is dominated by the transport properties of a narrow slab of coupled orbits in the geometry in whichJ ∥ [11\(\bar 2\)0] andH lies in the (11\(\bar 2\)0) plane. In samples whose quantum state lifetimeτs exceeds 10−10 sec the conductivity of the coupled orbit system is controlled byτs as well as the large-angle scattering lifetimeτ1. The magnetoresistance then exhibits complicated behavior, oscillating with |H| and varying rapidly with field orientation. The behavior withH aligned to within about 0.001° of [10\(\bar 1\)0] is due to manifold multiply connected open orbits, whereas forH tilted away from [10\(\bar 1\)0] by more than this the behavior is due to extended, multiply connected closed orbits. Detailed models for both these regimes are developed, taking into account the interference effects resulting from the feedback paths. Within these models the transport properties are calculated exactly, taking into account to all orders the amplitude propagation of wave packets within the network. Results are compared to experimental data and previous calculations, and the limitations of the model are discussed.

A new quantitative theory is developed to calculate from first principles the transverse magnetoresistance of Mg for the special case in whichH is parallel to a [10 $$\bar 1$$ 0]-type axis andJ is parallel to a [11 $$\bar 2$$ 0]-type axis. For this case, magnetic breakdown produces a multiply coupled network of interfering electron trajectories which generate large-amplitude quantum oscillations in the transverse magnetoresistance. It is shown that two distinct regimes of quantum transport exist for these oscillations and that this theory can be used to derive the electron quantum state lifetime τ from experimental data. The sensitivity of this effect to τ extends the experimentally accessible range by four orders of magnitude from about 10 −12 to 10 −8 sec. In addition, since the quantum interference oscillations are essentially insensitive to the temperature dependence of the Fermi-Dirac distribution function, this effect is ideally suited to study the quantum state lifetime dependence on electron-electron scattering.

We report here the results of an extensive investigation of the de Haas-van Alphen effect in white tin. These results agree qualitatively with the predictions of Weisz's pseudopotential model for the tin Fermi surface; for every case in which Weisz's model differs qualitatively from the single-orthogonalized-plane-wave model, our data agree with the former. Quantitative agreement, however, has not been obtained. The experimentally determined cross-sectional areas of the Fermi surface typically differ by about 30% from those calculated by Weisz. Area branches are also reported that are due to coupled orbits generated by magnetic breakdown of the spin-orbit-coupling-induced energy gap between the third and fourth bands near the $\mathrm{XL}$ and $\mathrm{XP}$ zone lines.

The Fermi surface of beryllium has been throughly investigated by means of the de Haas-van Alphen effect, and the frequencies, though substantially in agreement with earlier measurements, have been obtained to greater accuracy. This has made it possible to construct a nonlocal pseudopotential model for the Fermi surface, in which the Fourier coefficients of the crystal potential are treated as parameters to be evaluated by fitting to the experimental data. In this way, all the principal cross sections of the surface have been fitted to within 1%, using only five adjustable parameters. Comparison of calculated cyclotron masses for these cross sections with the masses determined from the temperature dependence of the dHvA amplitudes indicates that the mass is enhanced about 20% by many-body interactions neglected in the model. Finally, the pressure dependence of the Fermi surface has been calculated and compared with some available experimental measurements.

Using a very pure single crystal of magnesium we have observed large-amplitude oscillations in the transverse magnetoresistance which result from the direct interference of normal-state electron waves. The amplitudes of these interference oscillations appear to be independent of the $\mathrm{kT}$ broadening of the Fermi distribution. The primary result of our experiment shows that these waves are phase coherent at 1.2\ifmmode^\circ\else\textdegree\fi{}K over macroscopic distances of at least 0.01 mm; this corresponds to a quasiparticle lifetime of at least 2 \ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}11}$ sec.

Measurements of the magnetoresistance of high-purity single crystals of magnesium were made in order to determine the magnetic field directions for which open trajectories exist on the magnesium Fermi surface. Nine distinct series of open trajectories with net directions parallel to the basal plane of reciprocal space were observed. These trajectories have basic periodicities extending up to nine Brillouin zones and result from magnetic breakdown between the monster and cigar. The magnetic field directions for which these trajectories exist are shown to be in excellent agreement with the predictions of the Fermi-surface model proposed by Kimball, Stark, and Mueller. Measurements for the open trajectory parallel to the hexagonal axis in reciprocal space are also reported; these show a clear indication of the onset of magnetic breakdown across the spin-orbit gap separating the monster and the cap.

A detailed analysis is presented of the large-amplitude temperature-insensitive oscillations in the transverse magnetoresistivity of ultrapure Mg single crystals resulting from the direct interference of electron quantum states. A calculation of the relative harmonic content of these interference signals based on the transmission characteristics of a magnetic breakdown-generated interferometer is used to quantitatively study certain aspects of the electron states in Mg as well as details of magnetic breakdown. In particular, values of magnetic breakdown parametersH 0are determined without invoking the complexities of transport theory. An absolute lower limit for the electron quantum state lifetime of τ ≳ 0.5 nsec is obtained (forT=1.5° K), although a best fit to the data gives a value an order of magnitude larger, τ∼5 nsec, which corresponds to quantum phase coherence extending over a distance of 3 mm in these crystals. In addition, this work provides direct experimental verification of the π/2 phase difference between transmitted and reflected electron states at a magnetic breakdown junction. Comparison of the results of this experiment with previous work via an existing semiempirical band structure calculation demonstrates the complete consistency of these measurements with previous Fermi surface data.

The galvanomagnetic properties of several single-crystal specimens of high-purity zinc have been investigated. The results of this investigation are presented and a Fermi-surface topology consistent with the data is discussed. Many of the detailed features of the galvanomagnetic properties are found to result from the effects of magnetic breakdown. The open orbit parallel to the hexagonal axis is modified but not eliminated by magnetic breakdown at fields of about 17.5 kG. A "giant orbit" of the type previously observed in magnesium is found in zinc. This orbit results from magnetic breakdown across the energy gap near $K$ which separates the second-band hole sheet from portions of the third-band electron sheets. Magnetic breakdown across this same energy gap gives rise to discrete bands of open trajectories in the basal plane when the direction of the magnetic field is tilted away from the hexagonal axis in the ($10\overline{1}0$) and ($11\overline{2}0$) crystallographic planes. The transition region of the magnetic breakdown which gives rise to the giant orbit is investigated in detail, and the large quantum oscillations which have been previously observed in the transport properties of zinc are shown to be a transition-region phenomenon. The oscillations appear to result from an oscillatory breakdown probability which, in turn, arises from oscillations in the density of states associated with the Landau levels of the needle-shaped portion of the Fermi surface of zinc. An analysis of the detailed shape of the oscillations indicates that the Landau levels of the needle are split into discrete spin levels with a very large effective $g$ factor. The three values of the effective $g$ factor which are compatible with the present data are ${g}^{*}=90$, ${g}^{*}=180$, and ${g}^{*}=360$, but an investigation with higher magnetic-field strengths will allow a unique choice to be made. No evidence has yet been obtained that yields a direct confirmation of the quasi-particle states predicted by Pippard.

We report here the results of a detailed investigation of the de Haas---van Alphen effect in magnesium. The data were taken for the three major crystallographic planes in magnetic fields extending to 52 kG and at temperatures down to 0.31\ifmmode^\circ\else\textdegree\fi{}K. The results yield cross-sectional area branches for every sheet of the single-OPW-model Fermi surface (OPW=orthogonalized plane wave). Coupled orbits generated by magnetic breakdown are shown to be responsible for many of the observed area branches. The results are in good qualitative agreement with existing models for the magnesium Fermi surface.

Geometric resonances in the ultrasonic attenuation have been studied in magnesium for sound propagated in the three principle crystallographic directions. A total of twenty different geometric resonance branches have been obtained. The assignment of these branches to Fermi-surface calipers shows that the magnesium Fermi surface is much more free-electron-like than currently thought. Three of the first four local pseudopotential Fourier expansion coefficients are estimated from data to be 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ 1.0 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$, and 1.7 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ Ry for the absolute value of the [$10\overline{1}0$], [0002], and [$10\overline{1}1$] coefficients, respectively. A two orthogonalized-plane-wave model is given for the lens-shaped piece of the Fermi surface in the third Brillouin zone, and various physical properties of this model are calculated and compared with experimental data.

Extensive de Haas-van Alphen cross-sectional areas and magnetoacoustic calipers for the magnesium Fermi surface were used to generate local- and nonlocal-pseudopotential model Hamiltonians. Both models yielded a good representation of the Fermi surface. Small but significant deviations from experimental data could not be eliminated using the local model, but these were eliminated with the nonlocal model. The values of the local-pseudopotential Fourier coefficients obtained were ${U}_{10\overline{1}0}=+0.014$ Ry, ${U}_{0002}=+0.026$ Ry, ${U}_{10\overline{1}1}=+0.036$ Ry, and ${U}_{10\overline{1}2}=+0.058$ Ry. The variational parameters for the nonlocal pseudopotential were the Fourier coefficients of the self-consistent ion-core potential $V(\mathrm{r})$. The values obtained for these were ${V}_{10\overline{1}0}=\ensuremath{-}0.440$ Ry, ${V}_{0002}=\ensuremath{-}0.412$ Ry, ${V}_{10\overline{1}1}=\ensuremath{-}0.386$ Ry, and ${V}_{10\overline{1}2}=\ensuremath{-}0.299$ Ry.

One of the experimental manifestations of superconductivity that has not yet received a quantitative understanding is the electronic specific heat Ces in most elemental superconductors. Many of these exhibit significant deviations from the BCS prediction usually yielding a larger value of Ccs than that predicted for T«Tc. For Tc/T ≳ 2 the BCS model gives Ces(T)/ γ.Tc = 8.5 exp (−1.44 Tc/T ) for an isotropic (T = 0) energy gap ΒO = 1.76 kBTc. The slope of Ces (T) is 1.44 instead of 1.76 because of two significant contributions to the energy change with decreasing temperature: one of these is the condensation of further Cooper pairs into the ground state, while the other comes from the increase in the energy gap. It is traditional to interpret this type of deviation as arising from anisotropy in the energy gap Δ K . and estimates of this anisotropy, for simple metals, have been based on the Ces data. Although independent confirmation of this approach has been generally lacking, we will demonstrate in the following, that such confirmation exists for zinc, which shows strong deviations from the BCS behavior.