A Spectroscopic Determination of Scattering Lengths for Sodium Atom Collisions

We report a preliminary value for the zero magnetic field Na ²S(f = 1, m = − 1) + Na ²S(f = 1, m = − 1) scattering length, a_1,−1. This parameter describes the low-energy elastic two-body processes in a dilute gas of composite bosons and determines, to a large extent, the macroscopic wavefunction of a Bose condensate in a trap. Our scattering length is obtained from photoassociative spectroscopy with samples of uncondensed atoms. The temperature of the atoms is sufficiently low that contributions from the three lowest partial waves dominate the spectrum. The observed lineshapes for the purely long-range 0g− molecular state enable us to establish key features of the ground state scattering wavefunction. The fortuitous occurrence of a p-wave node near the deepest point (R_e = 72 a₀) of the 0g− potential curve is instrumental in determining a_1,−1 = (52 ± 5) a₀ and a_2,2 = (85 ± 3) a₀, where the latter is for a collision of two Na ²S(f = 2, m = 2) atoms.     

Magnetic Rotation in the A = 80 Region: M1 Bands in Heavy Rb Isotopes

We have studied the isotopes ⁸²Rb₄₅, ⁸³Rb₄₆, and ⁸⁴Rb₄₇ to search for magnetic rotation which is predicted in the tilted-axis cranking model for a certain mass region around A = 80. Excited states in these nuclei were populated via the reaction ¹¹B + ⁷⁶Ge with E = 50 MeV at the XTU tandem accelerator of the LNL Legnaro. Based on a γ-coincidence experiment using the spectrometer GASP we have found magnetic dipole bands in each studied nuclide. The regular M1 bands observed in the odd-odd nuclei ⁸²Rb and ⁸⁴Rb include B(M1)/B(E2) ratios decreasing smoothly with increasing spin in a range of 13⁻ ≤ J^π ≤ 16⁻. These bands are interpreted in the tilted-axis cranking model on the basis of four-quasiparticle configurations of the type π(fp)πg9/22νg9/2. This is the first evidence of magnetic rotation in the A ≈ 80 region. In contrast, the M1 sequences in the odd-even nucleus ⁸³Rb are not regular, and the B(M1)/B(E2) ratios show a pronounced staggering.     

Resolution Limits of Analyzers and Oscillatory Systems

This paper considers the resolution limits of those analyzers and oscillatory systems whose performance may be represented by a second-order differential equation. The “signal uncertainty” product Δf·Δt is shown to be controlled by the ability of a system to indicate changes in energy content. The discussion refers the functioning of the system to a signal space whose coordinates are energy, frequency, and time. In this signal space, the product of the resolution limits, U = (ΔE/E₀) (Δf/f₀) (Δt/T₀) is the volume of a region within which no change of state in the system may be observed. Whereas the area element Δf·Δt is freely deformable, no operations upon either Δf or Δt can further the reduction of the energy resolution limit. Thus U is irreducibly fixed by the limiting value of ΔE/E₀. By considering the effects of noise upon ΔE/E₀, and thus upon U, the paper demonstrates the rise of statistical features as signal-to-noise ratios decrease.Functional relationships derived from ΔE/E₀ and U are tabulated. These equations facilitate computation of the limits of observable changes of state in a system, and they provide guidance for the design of experiments to apportion the uncertainties of measurement of transient phenomena as advantageously as possible. A reference bibliography and appendices giving somewhat detailed proofs are included.The basis of this paper is the consideration that the indication of most instruments used in measurement represents either the storage of energy or the flow of power. The least changes that the instruments can indicate, therefore, are controlled by the smallest discernible change in energy storage or power flow.The subject of the resolution limits of measuring instruments in terms of the least amounts of frequency change and the least time interval in which a change may be detected have been treated by several authors, among whom one may cite as examples Gabor, Kharkevich, and Brillouin—and, while this paper was being revised, Pimonow.¹ The present author has also discussed this relationship for scanning analyzers, and indicated that there were circumstances in which limitations were introduced by the presence of a least discernible increment of power or energy.² These papers (ref. 2) are quoted, in addition to the prior work, by Pimonow.Gabor pointed out, by analogy to quantum theory, that there was a “quantum” of information that could be described by the product of differentials representing the least discriminable increment of frequency that could be observed in an increment of time. This relation arose from the application of the Fourier transform to relate an increment of time to its corresponding increment in the frequency domain. The product Δf ? Δt ≈ 1was defined by Gabor as the “Logon.” The fact that, as he says, the product is “of the order of unity” is a consequence of the particular normalization he used in computing the Fourier transform for Gaussian pulses. A similar relation was presented by Brillouin, but as he computed Δt in terms of the half-powers of brief, symmetrical pulses, he found a somewhat different normalization factor, and obtained Δf⋅Δt=14π.Kharkevich adopted a somewhat more general expression for this equation, also in terms of a normalized Fourier transform, by first writing Δf ? Δt = Aand computing A for pulses of various forms. He remarked that A might differ if other criteria were chosen, but related A only to the form of the signal. Further, he pointed out that A is independent of the, damping of the system.The studies by Gabor. Kharkevich, and Brillouin were all carried out for essentially noiseless systems. This paper, on the other hand, does not normalize for unit energy, but considers the energy storage and dissipation in systems whose performance may be described by a linear differential equation of the second order. Thus, by dealing with the energy stored as well as with the time and frequency we are able to study the response of a system to signals other than variously shaped pulses of unit energy, and to signals in noise as well as to noiseless systems.We can also consider, in this way, the case which has been omitted from the previous work: the response of the system which may in itself have a “least count”³ or inherent internal noise.We introduce the means for taking into account the presence of noise with a signal, internal noise in an analyzer, or the least discernible indication of the analyzer (which may be a step limitation—such as a digital step, or a reading limit) by discussing the limits of the analyzer’s performance as being fixed by the least change in energy storage, ΔE, that can be resolved under the circumstances of analysis. Several conditons may combine to fix the value of ΔE. For example, an analyzer having appreciable self-noise may be used to detect a signal in noise. As a rule, the internal and external noise sources in that case would be incoherent, and the sum of the noise energies stored in the analyzer from those sources would fix the value of ΔE.The system for which this discussion is carried out is a system whose working may be described by a linear differential equation of the second order. This behavior is common to many physical systems occurring in nature, and to many instruments used to observe natural phenomena. All of these systems share the same properties, because they are properties inherent in the differential equation that describes them. By virtue of the second-order term, they may be seen to be capable of storing oscillatory energy reversibly. They will respond with a sinusoidal output after excitation by shock or noise. Under sinusoidal excitation, they will respond selectively to excitation of various frequencies.Systems to be discussed in this paper are those in which the storage of energy occurs in the coordinates describing the system: these systems are described by a second-order differential equation with constant coefficients; i.e., the system parameters are not affected by the energy storage process. By invoking the Boltzmann-Ehrenfest adiabatic principle,⁴ it is also possible to apply many of these equations to systems in which energy is stored by a change of parameters. However, this matter has not been investigated in detail.Although most of the systems to which the second-order differential equation is applicable take part in time-varying phenomena, there is nothing inherent that restricts the equation to functions of time. Certain spatial distributions also may be described by the equation—such as the magnetization on magnetic tape, some types of optical images, and some diffraction effects. Thus, although this paper will deal with application of the equation to time-varying phenomena, its conclusions are also applicable, with a judicious choice of variables, to spatial distributions.For the sake of a coherent structure upon which to base this paper, we choose a mechanical system of inertia, M, dissipation (proportional to velocity) D, and coefficient of restitution, k. This system has a single degree of freedom, along the coordinate x, and its force-free behavior is given by solutions of the homogeneous equation: Mx¨+Dx¨+kx=0.When energy is stored in the system, it is dissipated at a mean rate which bears a constant relationship to the amount of energy stored. This constant is a function of the parameters of the system. In terms of dissipation, the constant is frequently expressed by the relative damping, γ, the ratio of the damping of the system to the critical damping for no oscillation. A reciprocal quantity, the “figure of merit,” Q, is commonly used in communication problems. These two constants are related through the equation: γ=12Q.Because we are more concerned here with storage than with dissipation, the quantity Q will be used in the discussion.The conventional definition for Q applies when the system is driven at its resonance frequency; at other frequencies the ratio of the storage of energy to the rate of dissipation depends upon the driving frequency. When the system is free of excitation, the conventional definition of Q again applies. This value of Q will be denoted as Q₀, to distinguish it from the more general definition of Q to be applied in the appendix. Defining as the natural frequency of the system the quantity f₀, which is the natural frequency of the system in the absence of damping, f0=12πkMand Q0=PeakenergystoredatthenaturalfrequencyEnergydissipatedpercycleatthenaturalperiodIn terms of the parameters of the system, this definition of Q₀ is equivalent to the ratio: Q0=2πf0MD=k2πf0D=kMD.The differential equations for the transient and driven response of the system can be expressed in terms of Q₀, D, and f₀. For the force-free equation: x¨2πf0+x˙Q0+2πf0x=0and, for the driven response of the system to a sinusoidal force of amplitude A and frequency f: x¨2πf0+x˙Q0+2πf0x=AQ0Dej2πft.Letting the ratio of the driving frequency to the natural undamped frequency be represented by ? = f/f₀we can write down the phase relation between the driving force and the resulting motion. The phase angle, θ, when the steady-state condition has been attained is given by θ=tan−1ϕQ0(1−ϕ2)and the energy stored in the steady-state condition is given by: Es=A22D⋅Q02πf0⋅1Q02(1−ϕ2)+ϕ2.The application of the Fourier transform can be considered to be tantamount to referring the behavior of the system to one or the other of two mutually perpendicular planes: The steady-state condition is described by the representation of the state of the system in the energy versus frequency plane and it of necessity deals with the steady state because the variable, time, is not involved. The transient behavior of the system is represented in the energy versus time plane. It is easier to understand that this is orthogonal to the frequency representation if we omit the normalization often used, of choosing the unit of time in terms of the natural period of the system. Nevertheless, it is more convenient to express the behavior of a system in terms relative to its natural parameters, and for the sake of simplicity of equations, much of the discussion will be related to the natural properties of the system.Three properties serve to specify the behavior of an analyzer described by a second-order differential equation: its undamped natural frequency, f₀, its figure of merit, Q₀, and its “least count,” ΔE, the least energy change that can be resolved by the system. The changes in energy content of the system may be thought of as taking place within the signal space bounded by the energy-time and energy-frequency planes, and the representation of the behavior of the system as a function of time or of frequency may be considered as projections upon the principal planes. Since the actual use of the system as an analyzer is never wholly steady-state or completely broadband, the actual process of analysis may be considered as taking place in some plane within the signal space bounded by the principal planes. Depending upon the information sought, the plane of the analyzer will be close to one or the other of the principal planes.Ordinarily, an analyzer indicates a running time average over the energy, E_s, stored in it. For steady-state signals the indication becomes proportional to the input power; for signals of very brief duration the analyzer responds ballistically and thus gives an indirect indication of energy. The limit of resolution is fixed by the least change in energy storage, ΔE, that can be resolved under the circumstances of analysis. In this discussion we shall be dealing with incremental ratios. As the differential will always be considered jointly with the total energy stored over the same time interval, it will be possible in general to discuss ratios of incremental power, ΔW, to input power, W₀, or incremental energy, ΔE, to energy stored, E_s, interchangeably.As a function of time, the building up and decay of the energy stored within the system are exponential processes. Therefore it proves convenient to describe the behavior of the system in terms of an exponential variable. Thus, to express the changes in energy storage, we choose an exponential coefficient, α. The energy resolution limit, ΔE/E₀, may be expressed in terms of α through the following definition: If the initial amount of energy stored in the analyzer is E₀, and the minimum change of energy that can be discerned is ΔE, then in terms of the exponential variable α the equivalent statement is that the energy content must decrease to an amount (e^−α) times its original value for the change to be at least equal to the minimum change discernible. Expressed as an equation, this limit is given by ΔE= (1 — e^−α)E₀ for energy, and in the many cases in which we are dealing with energy flow through the system, alternatively as ΔW = (1 — e^−α) W₀ for power. From the definition of α in terms of ΔE and E₀, it is evident that: α=−ln(1−ΔEE0).However, it is not along the time axis alone that α proves such a convenient function. Because of the close relation between exponential and angle functions, it also yields simple equations for the behavior of the system as a function of frequency. For the foregoing reasons, we shall describe the manner in which the independent variable, the energy increment ΔE, influences the resolution limits in frequency and time in terms of the variable α, and we will then return to consideration of what the equations describing these resolution limits mean in terms of ΔE. We will also discuss special types of noise conditions that may give rise to the irreducible increment that ΔE represents.The conditions under which α sets the resolution limit along the time axis are derived from considering the system to contain an amount of energy E₀ at time t = 0, and at that instant and for some time subsequent to that, to be free of any driving force. The resolution limit along the time axis is fixed by the least time interval, Δt, during which the system is capable of changing its energy storage by the factor, e^−α. This corresponds to dissipating at least the discernible energy increment, ΔE, during the time Δt. The resolution limit stated in terms of the natural period of the system, T₀, is thus given by Δt/T₀.When a system of this sort is used as an analyzer, the time interval over which the observation takes place must be of sufficient length for some change to be indicated. Thus the observation interval, Δτ, must equal or exceed the least time interval Δt. (In the previous papers cited in reference 2, the observation interval Δτ was used in place of the least time increment Δt. Use of the least time increment converts several previously found inequalities to equations.)It is a very close approximation (see appendix 1) to consider only the exponential factor in the decay of energy in the system. The oscillatory terms arising from the sinusoidal nature of the dissipation process are always relatively minor. Thus, regardless of the precise initial conditions, EE0|(t=Δt)=e−α≈exp(−2πQ0⋅ΔtT0)and, therefore, the resolution limit along the time axis, expressed in terms of the natural undamped period of the system is ΔtT0≈αQ02π.From this equation one can see that the ratio of α to Δt is twice the real coordinate of the poles of the system on a Nyquist diagram.The system is selective with respect to the sinusoidal frequency of the driving force which acts as a source for the energy it stores. This makes it suitable for the detection of sinusoidal frequencies falling within its range of response, or a group of similar systems with different f₀’s may be used in the analysis of the components of complex signals. For this purpose, the observation takes place in a plane approximating the energy-frequency plane. Along the frequency axis, one may speak of a frequency resolution limit in this sense: If one knows the natural frequency f₀ to which an analyzer is tuned, then a maximum indication of energy storage for a steady-state sinusoidal signal corresponds to a signal in the vicinity of f₀. Until the energy storage has changed by an amount in excess of ΔE or, when expressed as a relative proportion, a factor in excess of ΔE/E₀, the departure from maximum indication is not observable. The change in frequency required to produce this effect, Δf, corresponds to the frequency resolution limit Δf/f₀.As one can see from the equation for the energy stored in the steady-state condition, the response of the system to a sinusoidal frequency other than its natural undamped frequency is diminished to a fraction, F, of the maximum energy that would be stored at the natural frequency. Expressed in terms of the ratio of the frequency of the driving force to the natural undamped frequency of the system (f/f₀=ϕ), the fraction is F=1ϕ2+Q02(1−ϕ2)2.The frequency limits for the region Δf surrounding f₀ are found by solving for the condition e^−α=F. Solving for the upper and lower frequency limits, f_b and f_a, for which the energy stored in the resonating system is a fraction e^−α of the peak response yields the expression: fb2−fa2f02=2Q0(eα−1)+14Q02.And to a very close approximation, the frequency resolution limit comes out as: Δff0=(eα−1)Q0when Δf is defined as (f_b—f_a).The foregoing discussion may now be summarized in geometrical form by reference to a three-dimensional figure in signal space. The limits of resolution of an analyzer may be represented by an irreducible region in a three-dimensional space that must be exceeded before any information about a signal can be found. This space is shown in figure 1. As one can see from the equations, the figure of merit, Q₀, enters into the resolution limits for time and frequency in a complementary way. Thus, it determines the relative proportions between the resolution limits Δf/f₀ and Δt/T₀. One may therefore apportion the relative uncertainties in frequency and time to suit the requirements of an experiment whenever one can control the Q of a system. This process is discussed more fully in “Uniform Transient Error” (see footnote ²). But the product of the resolution limits, the “uncertainty equation” depends in irreducible fashion upon α.Open in a separate windowFigure 1Resolution limits making up the indication limit, U.T₀ is the period corresponding to f₀, the natural undamped frequency of the system. E₀ is the maximum sinusoidal energy stored.Thus the signal uncertainty equation, expressed in terms of α, turns out to be Δf⋅Δt=12π(eα−1)where α is related to ΔE through the definition of the least change ΔE that may be observed in the total stored energy E₀.The basic form of the resolution equations results from substitution for the exponential coefficient, α, in the equations already derived: ΔtT0=−Q02πln(1−ΔEE0) ΔfF0=1Q0ΔEE0(1−ΔEE0) Δf⋅Δt=1πΔEE0⋅11−ΔEE0ln11−ΔEE0The volume corresponding to the irreducible limits in signal space, a quantity here defined as the “indication limit,” can be computed from the signal uncertainty equation. It is defined by the triple product of the resolution limits along the three axes, U=ΔEE0⋅ΔtT0⋅Δff0,and it is just the volume of the elementary figure shown in figure 1. Its computation might at first glance appear to be somewhat redundant to the signal uncertainty equation. In fact, its functional form permits factoring the expression for U into components that have an interesting connotation in physical measurements.Thus, since: U=ΔEE0⋅Δff0⋅ΔtT0≈ΔWW0⋅Δff0⋅ΔtT0the magnitude of U expressed in terms of the variable, α, is U=αe−α2π(eα−1)3/2and U is given in terms of the energy resolution limit, ΔE/E₀, by: U=12π⋅(ΔEE01−ΔEE0)3/2⋅[−(1−ΔEE0)ln(1−ΔEE0)].It should be noted that the indication limit U and the signal uncertainty relation Δf·Δt depend only upon the energy resolution limit. The energy resolution limit is an independent variable and cannot be reduced by operations altering the function of the system along the f and t axes: for instance, changing the Q₀ of the system.For a system operating at its optimum, the irreducible energy or power increment for the system would be set by the noise energy stored in the system. This noise energy may be due to Brownian motion in the system, for example. Noise of external origin may be present with the driving signal. In the general case the intrinsic and extrinsic noise powers will not be coherent, and the total noise energy stored in the system may usually be considered as the sum of contributions.The prior papers relating the resolution limits to the relative amounts of signal and noise present (see footnote ²) were based upon a derivation subject to the restriction that the signal-to-noise ratio be high. In those papers, the relationship found was Δf⋅Δτ≥(S/N)−3/22πwhere S and N were signal and noise powers, respectively, and where the use of the observation interval Δτ rather than the limiting time increment Δt made the relation an inequality.Such a restriction was, in fact, not required. A tractable and useful expression for the product Δf·Δt can be derived, valid for all values of S/N.In the case where both signal and noise energy are stored, the total energy present in the system is E₀=S+N. An increment in the signal energy stored (or in the input signal power) can be detected only if it equals or exceeds the minimum energy increment the system is capable of indicating. From this definition and the definition of the exponential factor, α: ΔSS+N≥ΔEE0=ΔWW0=(1−e−α).And, where the least energy increment is controlled by the noise energy stored; ΔE→N: or, very nearly ΔEE0=NS+N.Thus, where the limit of detection is set by the noise energy: α=lnN+SSfrom which the signal uncertainty becomes: Δf⋅Δt=12πNS·ln(1+NS).For high values of S/N, this expression approaches the limit (1/2π) (S/N) ^−3/2, a result found previously (see footnote ²).An especially interesting interpretation can be made from the form of the expression for the indication limit, U, when it is written in terms of the signal-to-noise ratio. U=(S/N)−3/22π⋅(−S/N1+S/NlnS/N1+S/N).The first factor can be recognized to be the expression for the signal uncertainty, Δf·Δt, when the signal-to-noise ratio is high. The term in parentheses also has a recognizable functional form, and in fact it is possible to relate it to the limiting probability of information transfer.To facilitate discussion, the factor N may be cleared from the fractions in the term, giving it the form: —S/(S+N) ln S/(S+N). As one can see from the series expansion for the natural logarithm, this product approaches the value N/S for large values of S relative to N, and the indication limit then is merely the trivial product of the reciprocal of the signal-to-noise ratio multiplied by the signal uncertainty function. It is quite another matter as S/N→0. Then the indication limit may be considered to consist of two meaningful terms: one is the same signal uncertainty function that has already been derived for high signal-to-noise ratios (see footnote ²) (i.e., for ΔE small re E₀); the other is a modulating function that we will now proceed to relate to statistical matters.Where one is dealing with the statistical presence of noise and signal, the long-time average signal-tonoise ratio may be described in terms of expected values. The following definition of expected value is taken from a textbook on statistics.⁵ “The expected value of a random variable or any function of a random variable is obtained by finding the average value of the function over all possible values of the variable…. This is the expected value, or mean value of x. It is clear that the same result would have been obtained had we merely multiplied all possible values of x by their probabilities and added the results … we might reasonably expect the average value of x in a great number of trials to be somewhere near the expected value of x.”A special case that is quite common experimentally is one in which the level distributions of signal and noise are precisely the same. If one has either a signal or noise, the probability of the signal being P, then it follows that the probability of the noise is (1–P). If the signal and noise have the same level distribution, G, the modulating term SN+SlnSN+S→P⋅GP⋅G+(1−P)⋅GlnP⋅GP⋅G+(1−P)⋅Gand thus the term that modulates the signal uncertainty function can be seen to reduce to the form —PlnP, where P is the probability of signal occurrence. From very simple considerations, therefore, the limit of detection is shown to be related directly here to a limiting probability of information transfer—a quantity usually derived in information theory by considering signals and noise to be made up of equal-sized unit impulses.It is interesting that this point was arrived at in the reverse direction by Woodward and Davies.⁶ They started with the PlnP term from Shannon’s information function and demonstrated from considering the signals and noise in radar detection that the quantity P was related to the signal-to-noise ratio for radar signals.However, the modulating term in its original form, in terms of S and N, may be seen to represent a generalization of the function defined in information theory as the channel capacity, H. This form is more closely related to the ordinary formulation describing the entropy of a system in terms of the probability distribution of energy states within it.⁷The question of whether a signal is detected in the presence of noise depends upon what the investigator chooses to consider a reasonable limiting probability in deciding whether a signal has been detected. A preponderance of only 1 percent above random distribution would correspond to a much smaller signal-to-noise ratio than would 90 percent. In fact, the common definition that gives the limit of detection as a signal-to-noise ratio of unity corresponds directly to setting the criterion for the limiting probability of detection at 50 percent.For the case N=S, the “—PlnP” term becomes ?½ ln ½ and the indication limit becomes: U1=12π(−12ln12)=ln24πfor which the signal uncertainty function is: Δf⋅Δt=ln24π.These are the limiting forms also where signals and noise are transmitted as “bits.”Thus far we have discussed noise from the standpoint of the noise energy stored in the system. This energy is, so far as the actual storage in the system is concerned, not distinguishable from energy stored that might be derived from purely sinusoidal excitation. If all excitation were withdrawn, and the system left in the force-free state, the energy in it would be dissipated in the usual exponential decay, and the oscillations during the decay would be essentially of frequency f₀, providing the system had moderate energy storage capacity (Q₀>1).Therefore, unless we have some other means for distinguishing among the sources of the energy stored in the system—such as, for example, knowledge of the spectral character of pulsed signals applied to the system—or knowledge of the amount of energy found present in the system when it is considered to be free of any known source, we are left to regard as signal that part of the energy stored in the system that was produced by a sinusoidal signal of power S_i. The remaining driving sources, of more or less broad spectral distribution, would in general be classified as noise.If the system is being driven at its natural frequency, its output signal will be Q₀/2π times its input, for Q₀ is 2π times the ratio of the energy stored to the energy dissipated during the cycle. At any other driving frequency, the energy stored will be weighted by the response, ρ, which relates the energy stored to the power supplied to the system. ρ=Q0/2πf0f2/f02+Q02(1−f2/f02)2.Suppose the noise within the system arises from a source whose spectral distribution is given by the power density function, N_f. The system will store energy with a weighting factor of ρ.The total energy stored in the system due to excitation by noise will then be found from the integral: N=∫0∞Nfρdω.Although we have discussed the behavior of the system in terms of cycles per second, the quantity we have designated as Q₀ is defined directly in terms of the ratio of the energy stored to the energy dissipated per radian. As the parameter of the function we are integrating is Q₀, we must choose the dimensionally similar variable in order to carry out the integration correctly. Thus we must integrate with respect to ω rather than f. This point is given in detail because it is an instance of the dimensionality of angles recently pointed out by C. H. Page.⁸This integral may be evaluated easily for noise of constant energy per unit bandwidth in cycles per second; the result is then⁹ N=πNf2and it is independent of Q₀ because the energy storage capacity of the system and its bandpass for noise are affected in a complementary fashion by changes in the figure of merit.For several other types of noise, an approximate equivalent white noise coefficient, n_f, can be defined, for which the foregoing equations remain applicable. Given a noise whose spectral distribution is v(f), a mean value “equivalent” white noise per unit bandwidth may be computed from the equation: nf=∫fafbv(f)df∫fafbdf.Obviously if v(f) equals a constant, then n_f is the familiar “White Noise” coefficient. However, for several other types of noise the mean-value integration yields an equivalent n_f, which may be treated as a constant, with rather low residual error resulting from this approach. This can be seen from the fact that the system is selective with respect to the frequency components of the power sources from which it stores energy. Thus the restriction is only that the spectral distribution be changing slowly in the frequency region immediately surrounding f₀. For the following noise distributions, the residual terms discarded amount, in the worst case, to no more than 12½ percent of the approximate value:For the noise distributions shown in the left-hand column, the respective mean values are shown in the right-hand column: v(f)=k⋅fnf=k2(fb+fa) v(f)=k/f(constantenergy/octave)nf=klnfb/fa(fb−fa)where, for this purpose, f_b and f_a are taken as the upper and lower half-energy limits of the response weighting function, ρ.One can use the foregoing to predict the ratio of energy stored from the sinusoidal excitation to the energy stored from the source of noise. Thus the effective signal-to-noise ratio in terms of energy stored in the system is given by: S/N=2Siρπnff0.The signal-to-noise ratio will be a maximum if the frequency of the sinusoidal excitation is just equal to the natural undamped frequency of the system. (For this condition, ρ = Q₀/2π.) The maximum ratio is: S0/N=SiQ0π2nff0.The foregoing derivation has meaning also with respect to any system described by a second-order differential equation in which some event of brief duration occurs. A quantity directly analogous to the indication limit may be computed. In this instance, the expressions relate the resolution limits that bound the occurrence of the event, specifying the least increments of energy, frequency and time within which the event can take place.Further, the limiting energy increment, ΔE, need not be formed by noise. For example, in a pulse- height system it would correspond to the smallest step in pulse height that might be indicated. As another example, an atomic system described by a second-order differential equation might have ΔE subject to quantum limitations.     

A Quantum Algorithm Detecting Concentrated Maps

We consider an arbitrary mapping f: {0, …, N − 1} → {0, …, N − 1} for N = 2ⁿ, n some number of quantum bits. Using N calls to a classical oracle evaluating f(x) and an N-bit memory, it is possible to determine whether f(x) is one-to-one. For some radian angle 0 ≤ θ ≤ π/2, we say f(x) is θ − concentrated if and only if e^2πif(x)/N ? e^{i[ψ₀?θ,ψ₀+θ]} for some given ψ₀ and any 0 ≤ x ≤ N − 1. We present a quantum algorithm that distinguishes a θ-concentrated f(x) from a one-to-one f(x) in O(1) calls to a quantum oracle function U_f with high probability. For 0 < θ < 0.3301 rad, the quantum algorithm outperforms random (classical) evaluation of the function testing for dispersed values (on average). Maximal outperformance occurs at θ=12sin−11π≈0.1620 rad.     

Dissociation Constant of 4-Aminopyridinium Ion in Water From 0 to 50 °C and Related Thermodynamic Quantities

The dissociation constant of 4-aminopyridinium ion in water at 11 temperatures from 0° to 50° C has been determined from electromotive force measurements of 19 approximately equimolal aqueous buffer solutions of 4-aminopyridine and 4-aminopyridinium chloride. Cells without liquid junction were used; the cell is represented as follows: Pt; H₂(g), H₂NC₅H₄N ? HCl(m₁), H₂NC₅H₄N(m₂), AgCl; Agwhere m is molality.Between 0° and 50° C, the dissociation constant (K_bh) is given as a function of temperature (T) in degrees Kelvin by −logKbh=2575.8T+0.08277+0.0013093TThe changes of Gibbs free energy (ΔG°), of enthalpy (ΔH°), of entropy (ΔS°), and of heat capacity (ΔC_p°) for the dissociation process in the standard state were calculated from the constants of this equation. At 25° C the following values were found: −logKbh=9.114,ΔG°=52,013jmole−1,ΔH°=47,090jmole−1,ΔS°=−16.5jdeg−1mole−1,ΔCp°=−15jdeg−1mole−1.Thermodynamic constants for the basic dissociation of 4-aminopyridine at 25° C were also computed.     

Thermodynamic Properties of Magnesium Oxide and Beryllium Oxide from 298 to 1,200 °K

As a step in developing new standards of high-temperature heat capacity and in determining accurate thermodynamic data for simple substances, the enthalpy (heat content) relative to 273 °K, of high purity fused magnesium oxide, MgO, and of sintered beryllium oxide, BeO, was measured up to 1,173 °K. A Bunsen ice calorimeter and the drop method were used. The two samples of BeO measured had surface-to-volume ratios differing by a factor of 15 or 20, yet agreed with each other closely enough to preclude appreciable error attributable to the considerable surface area. The enthalpies found for MgO are several percent higher than most previously reported values. The values are represented within their uncertainty (estimated to average ± 0.25%) by the following empirical equations³ (cal mole⁻¹ at T °K) MgO:HT°−H273.15°=10.7409T+1.2177(10−3)T2−2.3183(10−7)T3+2.26151(105)T−1−3847.94.BeO:HT°−H273.15°=11.1084T+7.1245(10−4)T2+8.40705(105)T−1−5.31245(107)T−2−5453.21.Values of enthalpy, heat capacity, entropy, and Gibbs free-energy function are tabulated from 298.15 to 1,200 °K.     

Acid Dissociation Constant and Related Thermodynamic Quantities for Triethanolammonium Ion in Water From 0 to 50°C

Earlier studies of the dissociation constants of monoethanolammonium and diethanolammonium ions and the thermodynamic constants for the dissociation processes have been supplemented by a similar study of triethanolammonium ion from 0° to 50° C. The dissociation constant (K_bh) is given by the formula ?log K_bh = 1341.16/T + 4.6252 ? 0.0045666Twhere T is in degrees Kelvin. The order of acidic strengths of the ions is as follows: Triethanolammonium >diethanolammonium>monethanolammonium. Conversely, monoethanolamine is the strongest of the three bases. The thermodynamic constants for the dissociation of one mole of triethanolammonium ion in the standard state at 25° C are as follows: Heat content change (ΔH°) 33.450 joule mole⁻¹; entropy change (ΔS°), −36.4 joule deg⁻¹ mole⁻¹; heat-capacity change (ΔCp°), 52 joule deg⁻¹ mole⁻¹.     

A Preliminary List of Levels and g-Values for the First Spectrum of Thorium (Th I)

The present state of the analysis of the first spectrum of thorium (Th I) is discussed briefly. Even and odd levels are listed in ​and2.2. The low even levels form terms arising from the configurations 6d² 7s² and 6d³ 7s. The Th I standard wavelengths that fit into the known level arrays are presented in

The Spectrum of Singly Ionized Atomic Iodine (I ii)

The I ii spectrum has been excited in electrodeless lamps and photographed from 655 A to 11084 A. Wavelengths and estimated intensities are given for almost 2,400 lines. A revision and extension of the earlier analyses of this spectrum has increased the number of known even levels from 43 to 124, and the number of odd levels from 55 to 190. New g_J-factors are given for 46 levels, and the previous designations of 40 levels are changed. Improved measurements in the vacuum ultraviolet region give a correction of 7.4 cm⁻¹ to be subtracted from the values listed in Atomic Energy Levels, Vol. 3 (1958), for all levels above the ground configuration. The approximately 1,800 classified lines now include all of the strongest lines. The ¹S₀ of the ground configuration 5s²5p⁴ has been found, and this configuration has been fitted to intermediate coupling theory. Magnetic dipole transitions between levels of the ground configuration, ³P₂–¹D₂ (7282 A) and ³P₁–¹S₀ (4460 A), have been observed and their nature confirmed by the Zeeman effect. The line 5p^{4 3}P₂–¹D₂ shows hyperfine structure which is in approximate agreement with a theoretical calculation of the expected structure. New levels have been found for almost all higher configurations. All previously known series have been extended and new ones found. From one of the new series, 5p3(4S°)5−12g5G6°, the principal ionization energy for I ii (154304 ±1 cm⁻¹) has been derived. The results of the analysis are compared with theoretical expectations in a number of cases.     

Heat of Formation of Calcium Aluminate Monosulfate at 25 °C

The heat of formation of calcium aluminate monosulfate, 3CaO·Al₂O₃·CaSO₄·12H₂O, at 25 °C, and of less completely hydrated samples of the same compound, was determined by the heat-of-solution method, with 2N HCl as the solvent, and 3CaO·Al₂O₃·6H₂O(c) and CaSO₄·2H₂O(c), as the reactants. The results were as follows:

Vapor Pressures of Ruthenium and Osmium

The vapor pressures and heats of sublimation of ruthenium and osmium have been measured using a microbalance technique based on the Langmuir method. Heats of sublimation at 298 °K were calculated with the aid of free energy functions. Least squares lines for the vapor pressure data, heats of sublimation, and normal boiling points were obtained as follows :

1. Ruthenium: LogPatm=7.500−32,769T(1940−2377°K) ΔHs°(298)=156.1±1.5kcal/molebp=4150±100°K
2. Osmium: LogPatm=7.484−39,880T(2157−2592°K) ΔHs°(298)=189.0±1.4kcal/molebp=5300±100°K

The indicated uncertainties are estimates of the overall limits of error.     

Drag force acting on a neuromast in the fish lateral line trunk canal. II. Analytical modelling of parameter dependencies

In Part I of this two-part study, the coupled flows external and internal to the fish lateral line trunk canal were consecutively calculated by solving the Navier–Stokes (N–S) equations numerically in each domain. With the external flow known, the solution for the internal flow was obtained using a parallelepiped to simulate the neuromast cupula present between a pair of consecutive pores, allowing the calculation of the drag force acting on the neuromast cupula. While physically rigorous and accurate, the numerical approach is tedious and inefficient since it does not readily reveal the parameter dependencies of the drag force. In Part II of this work we present an analytically based physical–mathematical model for rapidly calculating the drag force acting on a neuromast cupula. The cupula is well approximated as an immobile sphere located inside a tube-shaped canal segment of circular cross section containing a constant property fluid in a steady-periodic oscillating state of motion. The analytical expression derived for the dimensionless drag force is of the form |FN|/(|PL−PR|π(D/2)2)=f(d/D,Lt/D,ωD*), where |F_N| is the amplitude of the drag force; |P_L−P_R| is the amplitude of the pressure difference driving the flow in the interpore tube segment; d/D is the ratio of sphere diameter to tube diameter; L_t/D is the ratio of interpore tube segment length to tube diameter; and ωD*=ω(D/2)2/ν is the oscillating flow kinetic Reynolds number (a dimensionless frequency). Present results show that the dimensionless drag force amplitude increases with decreasing L_t/D and maximizes in the range 0.65≤d/D≤0.85, depending on the values of L_t/D and ωD*. It is also found that in the biologically relevant range of dimensionless frequencies 1≤ωD*≤20 and segment lengths 4≤L_t/D≤16, the sphere tube (neuromast–canal) system acts as a low-pass filter for values d/D≤0.75, approximately. For larger values of d/D the system is equally sensitive to all frequencies, but the drag force is significantly decreased. Comparisons with N–S calculations of the drag force show good agreement with the analytical model results. By revealing the parameter dependencies of the drag force, the model serves to guide biological understanding and the optimized design of corresponding bioinspired artificial sensors.     

Heat of Combustion of Borazine,B3N3H6

The heat of combustion of liquid borazine has been determined according to the following equation: B3N3H6(liq)+15/4O2(g)+3/2H2O(liq)=3H3BO3(c)+3/2N2(g),ΔH°(25°C)=−2313.3±12.6kj/mole(−552.90±3.0kcal/mole).From this value the heat of formation of liquid borazine may be calculated as ΔHf°(25 °C) = −548.5 ± 13.4 kj/mole (−131.1 ± 3.2 kcal/mole), and for the gas, ΔHf°(25 °C) = −519.2 ± 13.4 kj/mole (−124.1 ± 3.2 kcal/mole).     

Infrared Studies in the 1- to 15-Micron Region to 30,000 Atmospheres

A pressure cell was constructed using a pair of type II diamonds for study of infrared spectra of solids in the 1- to 15-micron region. Using commercial infrared equipment, spectra can be studied routinely to calculated pressures as high as 30,000 atmospheres. Under pressure, bands generally shift to higher frequencies and decrease in intensity. The magnitude of both changes depends on the mode of vibration. Occasionally major changes in spectra occur. In calcite the carbon-oxygen symmetric stretching, mode v₁, becomes active at elevated pressures while the doubly degenerate v₃, stretching, and v₄, bending, frequencies split. From the shift in frequency of v₁ with pressure the “compressibility”, [(−1/R_o) (dR/dp)], of the C—O bond length, R, is calculated to be 2.8×10⁻⁷/atmosphere. Major spectral changes are not observed in the same pressure ranges in other carbonates having the calcite or aragonite structures. The results for calcite may be explained by a shift of the CO3= ion from the trigonal axis under pressure.     

Heats of Hydrolysis and Formation of Dimethoxychloroborane

The heat of hydrolysis of dimethoxychloroborane has been measured; for the reaction, (CH3O)2BCl(liq)+3H2O(liq)=H3BO3(c)+2CH3OH(liq)+HCl(g)ΔH(25°C)=−26.6±0.8kj/mole=−6.4±0.2kcal/mole.From this, we have calculated the heat of formation of dimethoxychloroborane: for the liquid, ΔHf° (25 °C) = −782.1 ± 1.8 kj/mole (−186.9±0.4 kcal/mole), and for the gas, ΔHf° (25 °C) = −747.9 ±2.2 kj/mole (−178.8±0.5 kcal/mole).     

Heat of Formation of Titanium Tetraiodide

The heat of formation of titanium tetraiodide was determined relative to that of titanium tetrabromide by comparison of their heats of hydrolysis in dilute sulfuric acid. The difference in the heats of formation may be expressed by the equation: TiI4(c)+2Br2(liq)=TiBr(c)+2I2(c),ΔH(25°C)=−230.91±0.75kj/mole(−55.19±0.18kcal/mole).By taking the heat of formation of TiBr₄(c) as −616.72 ±4.60 kj/mole, the heat of formation of TiI₄(c) is calculated to be −385.81 ±4.64 kj/mole (−91.21 ±1.11 kcal/mole). The heats of hydrolysis of TiBr₄ and TiCl₄ were similarly measured; the value obtained for the difference (186.77 ±1.34 kj/mole) is in good agreement with the difference between the directly determined heats of formation (187.11 ±5.35 kj/mole).     

Heat of Formation of Titanium Trichloride

A calorimetric comparison of the heat of hydrolysis of TiCl₄(liq) with the heat of oxidation and hydrolysis of TiCl₃(c) has been made. The following value is reported for the combination of these data according to the process: TiCl3(c)+½I2(c)+HCl(g)→TiCl4(liq)+HI(g),ΔH°(25°C)=8.37±0.30kcal/mole.A combination of this value with −192.3 ±0.7 kcal/mole for the heat of formation of TiCl₄(liq) and with the standard heats of formation of HCl(g) and HI(g) gives for TiCl₃(c), ΔHf°(25 °C) = −172.4±0.8 kcal/mole.     

Precipitation and Solubility of Calcium Hydrogenurate Hexahydrate

Solid phases formed in the quaternary system: uric acid—calcium hydroxide —hydrochloric acid—water aged for 2 months at 310 K were studied to determine conditions for calcium hydrogenurate hexahydrate, Ca(C₅H₃N₄O)₂ · 6H₂O precipitation. The precipitates were identified by chemical and thermogravimetric analyses, x-ray powder diffraction, infrared spectroscopy, light microscopy, and scanning electron microscopy. In the precipitation diagram the concentration region in which calcium hydrogenurate hexahydrate precipitated as a single solid phase was established.The solubility of calcium hydrogenurate hexahydrate was investigated in the pH range from 6.2 to 10.1 at different temperatures. The total soluble and ionic concentration of calcium (atomic absorption spectroscopy and Ca-selective electrode), total urate concentration (spectrophotometry), and pH were determined in equilibrated solutions. The data are presented in the form of tables and chemical potential diagrams. By using these data the thermodynamic solubility products of calcium hydrogenurate hexahydrate, K_s = a(Ca²⁺) · a²(C₅H₃N₄O₃⁻), were determined: pKs=10.12±0.07at288K,pKs=9.81±0.09at298K,pKs=9.28±0.04at310K,andpKs=9.01±0.03at318K.The formation of calcium hydrogenurate hexahydrate crystals in urinary tract of patients with pathologically high concentrations of calcium and urates (hypercalciuria and hyperuricosiuria) is possible.