Average Step Size Research Articles

Unimolecular processes in vibrationally highly excited cycloheptatrienes. II. Steady-state photoisomerization

The steady-state unimolecular photoisomerizations of cycloheptatriene, 7-methyl-cycloheptatriene, and 7-ethyl-cycloheptatriene have been studied at wavelengths between 238 and 289 nm. The reactions proceed via vibrationally highly excited electronic ground state molecules. The specific rate constants k (E) for the unimolecular aromatizations and H-atom shifts are derived and compared with statistical calculations. Absolute values of the average step sizes <ΔE≳ for energy transfer of highly excited molecules are obtained via calibration of Stern–Volmer curves against specific rate constants from thermal isomerization experiments.

Application of multistep deactivation processes in the interpretation of intermolecular energy transfer following chemical activation by kinetic techniques

Interpretation of vibrational energy transfer following kinetically controlled chemical activation is refined by incorporating multistep deactivation processes into the RRKM treatment of the excited molecule. The functional form of the initial primary product energy distribution used is based on that suggested by Bunker. This model is applied in interpreting collisional energy transfer from cyclobutane-t, chemically activated by nuclear recoil reaction. New low pressure experimental data are used to estimate the average energy of the nascent cyclobutane-t and the average step sizes for energy transfer to He, N/sub 2/, CO/sub 2/, and parent on collision based on a stepladder deactivation model. Step sizes found for cyclobutane, He, N/sub 2/, and CO/sub 2/ are 10.0, 0.5, 2.0, and 5.0 kcal, respectively. 4 figures, 3 tables.

Intermediate structure effects in the reaction 23Na(p, α) 20Ne

Excitation functions of the reaction 23Na(p, α) 20Ne have been taken at eight angles between θ L = 30°and170° for proton energies E p = 4.0–8.0MeV. The average step size was Δ E = 12keV. An autocorrelation analysis of the strongly fluctuating α 0 yield curves revealed the presence of two coherence widths at a mean excitation energy of E x( 24Mg) = 17.5MeV. The smaller width Γ = 58±4keV was the only one observed in a simultaneously recorded α 1 yield curve and is identified with the fine structure or compound nuclear width. No indications of a J-or T-dependence were found. The larger width of about 150 keV is interpreted in terms of intermediate structure effects. A comparison with autocorrelation analyses of several hundred synthetic excitation functions was performed.

Temperature dependence of vibrational energy transfer in the “low” pressure competitive two-channel 1,1-cyclopropane- d2 system

Intermolecular energy transfer has been studied in the two-channel competitive isomerization of 1,1-cyclopropane- d 2, both in the neat system and in the presence of helium bath gas at values of k/k − centered around 0.02. The competing path ways differ in threshold energy by ≈ 0.6 kcal. The temperature range 773 K to 973 K was covered. Several methods of treating the data, whether by isotopic ratios of rate constants or by temperature dependence of fall off, are each independent of a knowledge of collision cross sections. Used in conjunction, they provide measurements of these quantities. Cyclopropane is an operationally strong collider (β ω = 1) for itself at 773 K with an average down step size, <Δ E/s> >/ 10 kcal mole −1 (>/ 3500 cm −1). At 973 K the substrate is no longer a strong collider; β ω declines to ≈ 0.55 with <Δ E/s> ≈ 5.2 kcal mole −1. For helium the corresponding quantities are β ω ≈ 0.078 <Δ E/s> ≈ 1.1 kcal mole −1 declining to β ω ≈ 0.010 and <Δ E/s> ≈ 0.53 kcal mole −1. The several methods of measuring these quantities give excellent independent agreement. Comparison with the earlier theoretical formulation of Tardy and Rabinovitch gives good agreement, the temperature dependence of β ω for the weak collider, helium, follows the relation β ω ∝ T − m , where m /s> 2.

Estimation of the Signal-to-Noise Ratio in High Information and Constant Factor Delta Modulation Systems

A "statistical average step size," γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> , which is a function of the rms power of a band-limited white Gaussian input signal, is defined for high information and first order constant factor delta modulation systems. The signal-to-noise ratios (SNR) are then calculated for these instantaneously adaptive delta modulation systems by substituting γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> for the fixed step size in the formulas for quantization (i.e., granular) and slope-overload noise applicable to linear (single integration) delta modulation systems. The calculated SNRs are in close agreement to those obtained by computer simulations. As γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> is difficult to calculate exactly, an empirical expression is found which is applicable to different encoding conditions. This empirically formulated γ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">av</inf> can be used to estimate SNR to an acceptable accuracy for most practical situations. A noise component not present in linear delta modulation is identified and called "overshoot noise" and the cross-correlation ρ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xe</inf> between input and error signals is studied. The SNR and ρ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xe</inf> for linear delta modulation are presented as reference bases.

Idle-Channel Noise Characteristic of a Delta Modulator with Step Imbalance and Quantizer Hysteresis

In 1969 Iwersen [4] analyzed the quantizing noise (assuming no slope overload) associated with a delta modulator ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\DeltaM</tex> ) with step-size imbalance. In this concise paper we consider the effects of both step-size imbalance and quantizer threshold hysteresis on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\DeltaM</tex> quantizing noise. The analysis is restricted to the important case of zero input signal. For this case we extend Iwersen's work to show that quantizer threshold hysteresis can significantly change the spectral shape of the idle channel noise. Our approach is to first construct two locus functions, one for the samples (of the steady-state noise waveform) taken at arbitrarily defined even sampling times and one for those taken at the odd sampling times. The noise waveform is then obtained as a superposition of interleaved samples of the even and odd loci. It is shown that the noise spectrum has a discrete tonal struture whose frequencies are independent of hysteresis and are determined solely by step imbalance as in Iwersen's analysis. Hysteresis changes the magnitude and phase of the discrete tones. We characterize the tone magnitude changes due to hysteresis and show that hysteresis can amplify, attenuate, or completely eliminate a tone. We present curves which specify the tone magnitude attenuation as a function of step imbalance ε, average step size σ, and hysteresis <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> . We also calculate P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> , the total noise power in the baseband up to half the sampling frequency. It is shown that for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 \leq h \leq \epsilon</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{T} = \sigma^{2}/3</tex> independent of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> and ε ;for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h &gt; \epsilon</tex> , P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> is an increasing function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon/\sigma</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h/\epsilon</tex> .

A central limit theorem for families of stochastic processes indexed by a small average step size parameter, and some applications to learning models

Let ϑ > 0 be a measure of the average step size of a stochastic process {p n (θ) }n=1(θ). Conditions are given under whichp n (θ) is approximately normally distributed whenn is large and ϑ is small. This result is applied to a number of learning models where ϑ is a learning rate parameter andp n (θ) is the probability that the subject makes a certain response on thenth experimental trial. Both linear and stimulus sampling models are considered.

Collisional Energy Transfer in Thermal Unimolecular Systems. Dilution Effects and Falloff Region

To assess further the operational meaning of efficiency, collisional activation-deactivation efficiencies β for thermal unimolecular reactions in the falloff region have been computed on a stochastic model with use of an iteration procedure. Stepladder and exponential collisional transition probability models were used; detailed balance and completeness were obeyed. Computations were made for the methyl isocyanide, nitryl chloride, and cyclopropane systems for which accurate values of the microscopic specific rate probabilities kE were also obtained. For a particular collisional model, the efficiency was calculated as a function of the average step size, temperature, the order of reaction, and the dilution of substrate by heat bath molecules. For the conventional definition of efficiency, β decreases as the pressure increases. The quasiuniversal efficiency-step-size relations found earlier in the second-order region have been extended. Correlations with several systems studied in the falloff are made. The nature of the dilution effect on β noted previously for the low-pressure regime [D. C. Tardy and B. S. Rabinovitch, J. Chem. Phys. 45, 3720(1966)] is further amplified, and examined in the falloff region. The resolution of β into population and transition probability factors is also extended. The comparative behavior of the Poisson and exponential factors relative to the stepladder values, when various amounts of energy are transferred on the average, is illustrated. Experimental β values in the low-pressure region for the nitryl chloride and methyl isocyanide systems are correlated with conclusions derived from the calculations.

Collisional Transition Probabilities for Deactivation of Highly Vibrationally Excited Cyclopropane and Dimethylcyclopropane

Collisional deactivation of vibrationally excited cyclopropane and dimethylcyclopropane molecules, in their ground electronic states, has been studied with ethylene and cis-butene as the respective deactivators. Activation is to an energy region just above 100 kcal mole—1, which is reached by chemical activation. The activated molecules may isomerize, and the measurable cascade is to a reaction threshold at 60–65 kcal; below the threshold the excited molecules are stable with respect to reaction. The principle and characteristics of this steady-state method has been described in detail in Ref. 5. The two substrate molecules offer an interesting difference in internal rotational degrees of freedom. Cyclopropane was investigated at 298°, 598°, and 723°K, and dimethylcyclopropane at 573° and 673°K. The observed behavior was compared with model calculations for various assumed functional forms of collisional transition probabilities pij which include stepladder, Gaussian, and skewed distributions. The data, unfortunately, are too inaccurate to distinguish between these models clearly; but they do show that ethylene removes energy from cyclopropane in average steps which are at least 7 kcal (and in one system, perhaps, as high as 30 kcal); butene removes energy from dimethylcyclopropane in steps of at least 10–14 kcal; there is an indication that a step or Gaussian distribution fits the data better than a skewed exponential model. There is some evidence that the average step size decreases with increase of ambient temperature.