Showing papers on "Saturation (graph theory) published in 1977"

Cosmological upper bound on heavy-neutrino lifetimes

Abstract: An upper bound on the lifetime of a massive, neutral, weakly interacting lepton, ${\ensuremath{ u}}_{H}$, is derived from standard big-bang cosmology. Saturation of the bound and reasonable assumptions about the weak interaction of the ${\ensuremath{ u}}_{H}$ then yield a prediction of approximately 10 MeV for its mass.

Experimental study of power broadening in a two-level atom

Abstract: Two frequency-stabilized cw dye lasers and an atomic beam were used to prepare sodium atoms in the aligned $F=2$, ${M}_{F}=2$ ground state, and to study power broadening in the transition from this ground state to the $3{P}_{\frac{3}{2}}$ $F=3$, ${M}_{F}=3$ excited state. This system is, to extremely good accuracy, a two-level atom. The line shape was determined as a function of the incident field strength and found to be accurately a Lorentzian over a large range of incident field strengths. The power broadening and saturation at resonance of the two-level system also agree with theory.

Coupled substitutions in the tourmaline group

Abstract: Statistical analysis of 136 natural tourmaline compositions from the literature reveals the presence and extent of coupled substitutions involving several cations and structural sites. In schorls and dravites these are a dehydroxylation type substitution (1) (OH)−+R2+ = R3++O2− and an alkali-defect type substitution (2) R++R2+ = R3++□, Al3+ being the predominant R3+ action. Substitution (1) which represents solid solution towards a proton-deficient end-member, R+ R 3 3+ R 6 3+ (BO3)3 Si6O18O3(OH), accounts for three times as much of the observed compositional variability as does (2) which represents substitution toward a hypothetical alkali-free end-member, □(R 2 2+ R3+) R 6 3+ (BO3)3Si6O18(OH)4. The occurrence of both of these substituions produces intermediates between end-member schorl/ dravite, R+ R 3 2+ R 6 3+ (BO3)3Si6O18(OH)4, and a new series within the tourmaline group, R 1−x + R 3 3+ R 6 3+ (BO3)3Si6O18O3−x (OH)1+x. In addition to dehydroxylation type, 2(OH)−+Li+ = R3++202−, and possibly alkali-defect type, 2R++Li+ = R3++2□, substitutions, a third type Li++O2− = (OH)−+□, occurs in the elbaites giving rise to Li-poor, proton-rich species. All three substitutions serve to reduce the Li-content of natural elbaite which, as a result, does not attain the composition of the ideal end-member, Na(Li1.5Al1.5)Al6(BO3)3Si6O18(OH)4. Substitution from elbaite and schorl/dravite toward R 1−x + R 3 3+ R 6 3+ (BO3)3Si6O18O3−x(OH)1+x is very extensive and may be complete. Substitution toward R 1−x + R 3 3+ R 6 3+ (BO3)3Si6O18O3−x(OH)1+x results in improved local charge balance. The mean deviation $${\Delta \zeta \left( {\text{O}} \right)}$$ from oxygen charge saturation $$\left( {\zeta \left( {\text{O}} \right) = 2.0} \right)$$ is at a maximum in end-member schorl, dravite and elbaite. Substitutions (1) and (2) progressively decrease $${\Delta \zeta \left( {\text{O}} \right)}$$ but substitution (1) does so more effectively, which may explain its predominance in nature. However, alkali-defective end-members appear to be unstable regardless of $${\Delta \zeta \left( {\text{O}} \right)}$$ . Substitution (3) in the elbaites cannot be discussed on the basis of charge balance considerations at present due to the lack of structural information on proton-rich species.

High-dose neutron-irradiation effects in fcc metals at 4.6 K

Abstract: The rate of residual-resistivity increase and the isochronal recovery have been studied on the fcc metals Al, Ni, Cu, Pd, Ag, Pt, and Au irradiated at 4.6 K with reactor neutrons to a dose of about ${10}^{19}$ fast neutrons/${\mathrm{cm}}^{2}$. The rate of resistivity increase is nonlinear as a function of irradiation-induced resistivity; computer analysis shows that the data are best fitted with an expression having up to third-order terms in $\ensuremath{\Delta}\ensuremath{\rho}$. There are deviations from simple damage-rate theory in all cases, but an anomalous negative deviation from a linear law (convex curvature) is observed in Ni, Pd, Pt (and Fe). This behavior is most probably caused by a decrease of the specific Frenkel-defect resistivity due to defect clustering, an effect which should contribute in all metals after fast-neutron irradiation to high doses. Saturation values of resistivity and defect concentration as well as recombination volumes have been obtained more accurately than from previous work. The isochronal recovery is compared with previous lower-dose data. Stage I decreases and stage III increases with increasing neutron dose. After high-dose irradiation, correlated recovery in stage III becomes dominant in the case of Al, Cu, Ag, and Au.

Time-dependent projection-operator approach to master equations for coupled systems. II. Systems with correlations

Abstract: In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of $N$ molecules interacting via a two-body potential and a quantum-optical system of $N$ two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's ${F}_{1}(t)$ and ${F}_{2}(t)$. In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO $R(t)$, and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.

Nuclear magnetism of solid hydrogen at reduced ortho concentration

Abstract: The NMR of 200-ml STP of pure ${\mathrm{H}}_{2}$, frozen in a cavity resonating at 310 MHz, was studied as the sample aged for 400 h while cooled by a dilution refrigerator. The frequency separation between points of maximum and minimum slope of the NMR line decreased linearly with time for $24ltl240$ h at the rate of 180 Hz/h. At the highest ortho concentrations and the lowest temperatures, the line shape corresponded closely to the doublet expected for interacting randomly oriented dipole pairs broadened by intermolecular dipolar interactions. However, as the ortho concentration decreased below 0.5 the line became more square than this model would predict. At several times during the run, the temperature was raised and again lowered. At the lowest temperature, the line changed with time through shapes very similar to those found with increasing temperature at an earlier time. Evidence from the integrated intensity of the line suggests that the ortho-para conversion during the run increased relative to the rate expected for a random lattice with interacting pairs. Studies of recovery from selective saturation showed that cross relaxation within the NMR line creates a spin temperature in about 5s.

Nuclear magnetic resonance in LiF: A single nuclear-dipolar-spin temperature in crystals with more than one spin species

Abstract: In a crystal containing more than one species of nuclear spin in a large dc magnetic field, the secular dipolar interactions between all spins, unlike as well as like, form a single reservoir described by a single spin temperature. We demonstrated this concept with experiments on LiF. In particular, we showed that ${T}_{1D}$ of $^{7}\mathrm{Li}$ and $^{19}\mathrm{F}$ are equal. We also measured ${T}_{1}$ and ${T}_{1\ensuremath{\rho}}$ of $^{19}\mathrm{F}$ using $^{7}\mathrm{Li}$ detection. Finally we applied this concept to the Provotorov theory of saturation, which we extended to the case of a two-spin system, and demonstrated experimentally the validity of the treatment.

On the Individual Ergodic Theorem for Subsequences

Abstract: We define a concept of saturation for a sequence of integers $\{k_j\}$. In the main theorem we prove that if $\{k_j\}$ saturates and $T$ is any weakly mixing measure-preserving transformation on an arbitrary probability space, then there exists a dense set $\mathscr{D}_T \subset L^2$ such that for $f \in \mathscr{D}_T$ $$\lim_{N\rightarrow\infty} \frac{1}{N} \sum^N_{j=1} f(T^{k_j}x) = E(f) \mathrm{a.e.}$$ This has the following application to probability theory: Let $Y_1, Y_2,\cdots$ be independent and identically distributed positive (or negative) integer-valued random variables with $E(Y_1) < \infty$. Let $$k_j(\omega) = \sum^j_{l=1} Y_l(\omega) \quad j = 1,2,\cdots$$. Then there exists a set $C$ of probability one such that for $\omega \in C$ and for any weakly mixing measure preserving transformation $T$ on an arbitrary probability space $$\lim_{N\rightarrow\infty} \frac{1}{N} \sum^N_{j=1} f(T^{k_j(\omega)}x) = E(f) \mathrm{a.e.}$$ for all $f \in L^1$.