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1.
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2.
Using the velocity analyzer of Zartman with improved technique the combined velocity spectrum of Bi atoms and Bi2 molecules was obtained at 827°, 851°, 875°, 899°, 922°, 947° C. From the spectral distribution curves the relative abundance of Bi atoms and Bi2 molecules in the beams at the above temperatures could be determined to 1 per cent. The vapor pressure curve of Bi was obtained experimentally by the method of effusion and the values so obtained were combined with the degree of dissociation of the vapor as computed from the beams to give the heat of dissociation. The heat of dissociation was computed from the data, assuming the pressure to be given by the temperature of the crucible Tc. In calculating the heat of dissociation, the equilibrium temperature was taken as that of the slit chamber Ts which was 24° above Tc. The results of these calculations plotted with log10Kp as ordinates against 1Ts give a straight line whose slope yields the value of the heat of dissociation as 77,100±1200 calories. The curves for the distribution of velocities observed and computed on the assumption of a given ratio of Bi atoms to Bi2 molecules in the beam were compared in an attempt to test the law of distribution of velocities. On the high velocity side agreement in two curves was obtained within the limits of experimental accuracy. On the low velocity side important deviations were noted of such a sort that the observed curves below a velocity α2, (α is the most probable velocity) gave more molecules than the theory demanded. Other deviations were observed on some of the runs taken with a fourth slit in which a deficiency of molecules was observed between velocities of .75α and α2. This deviation was probably due to a warping of the fourth slit carriage due to heat. The nature of the variation at velocities less than α2 indicated the presence of molecules of greater mass than Bi2 in the beam and at the lower temperatures a distinct peak corresponding to Bi8 molecules was observed which were present to less than 2 per cent. The vapor pressure curve for Bi was determined by least square reduction of the observed points to be given by log10 P = ? 52.23 × 195.26T + 8.56 between 1100° and 1220° abs. It lies very close to the extrapolated curve given in the International Critical Tables.  相似文献   

3.
It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is mddt{v(I ? v2c2)12}. The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P?1
P={I?α1kddt + α2ddt(kddt)?·}?
where α1, α2, etc., are constants and k = (I ? v2c2)?12. The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ? in that equation by P(U) and P(?), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the ddt in P be now regarded as ??t + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.  相似文献   

4.
The resistance coefficient of a body moving in a fluid depends on Reynolds Number R, Mach Number M and the parameter gLU2, which is customarily neglected in view of small weight of the air. Here L denotes a characteristic length; U denotes the body's speed of translation. The author points that dimensional deduction of this parameter does not limit it to the acceleration of gravity, and that the resistance coefficient is affected by the general acceleration to which the air is subjected. Evaluation of the acceleration of the air flowing about spheres puts this parameter in the form LR, where the characteristic length L is interpreted as the mean free molecular path. Large and small spheres were found to have widely different values of the pressure coefficient Δpq for the same Reynolds Number or Mach Number. Here Δp denotes the difference in pressure between front stagnation point and the rear portion of the sphere, and q denotes the dynamic pressure. The plot of Δpq against the parameter LR removes this confusion. The low values of Δpq are found to be associated with LR below a certain critical value, and high values of Δpq with LR above the critical value, which apparently indicates the condition under which the flow separation takes place. Attention is called to the effect of air pressure on the separation as shown by the parameter LR, and its possible bearing on the drag in high altitude flying.  相似文献   

5.
The periodic differential equation (1+ε cos t)y&#x030B; + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |ε|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–ε plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to ε3. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±12, ±32, ±52. Three-timing is employed to extend the solution near k = ±12. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.  相似文献   

6.
A procedure is described for determining the characteristics of adiabatic flow through a rocket nozzle with and without composition change. The method of calculation is illustrated for the expansion of pure hydrogen gas from a chamber temperature of 306° K. and a pressure of 20.42 atm. to atmospheric pressure.The study indicates that the exhaust velocity and temperature are highest for flow where complete equilibrium is reached at each temperature with respect to the reaction
H2?2H
Flow with composition change requires a nozzle exit to nozzle throat area ratio somewhat greater than that determined for adiabatic flow without composition change for the same ratio of chamber pressure to exit pressure.The residence time in a given temperature range is computed as a function of gas temperature for the two types of flow. The results of this calculation may be used to determine the minimum required reaction rates which allow composition changes during flow through the nozzle.  相似文献   

7.
In the land-based radar direction of aircraft it was found necessary to transfer to a large plotting screen a 10X magnification of the PPI cathode-ray tube face. An instrument was developed to do this photographically, with the projected image lagging only 15 seconds behind the latest information. A single 360-degree sweep on the PPI tube is reduced to a 12-inch diameter image on Eastman Type 5302 Fine Grain Release Positive Film. The resultant image is rapid-processed by the application of hot solutions in a special processing cup which restricts the liquids to a small circular area on the film.The spent solutions are quickly removed by a vacuum line and the near dry image is indexed into the projection system where an air pressure gate completes the drying, cools the film, and holds it flat during projection.The projected image, at a magnification of over 300X, has sufficient quality to resolve all the pertinent detail of the tube face.The method is felt to have some general application.  相似文献   

8.
The paper describes the phenomena associated with the reflection of a sharply defined beam of hydrogen atoms from a crystal of LiF. Of primary interest is the fact that the atoms show interference effects in agreement with the wave mechanics theory and plane grating diffraction patterns are photographed. Evidence of the thermal agitation of the surface ions is obtained from the diffuse reflection with surrounds the specular beam.The Schrödinger wave equation for the motion of a free particle of mass m is
▽2ψ ? 4πmih?ψ?t = 0 (I)
. The solution of this equation corresponding to the kinetic energy mv22 is
ψ = Ae2πi(vt?σxx?σyy?σxz), (2)
where
v  mv22and σ mvh
. The motion of such a particle should have the characteristics of a plane wave of frequency ν and wave-length λ = 1σ. The experiments of various investigators1 have shown the validity of the wave theory of the motion of the free electron and have given values of the wave-length in agreement with the theory.The free motion of atoms, ions and molecules should likewise have wave characteristics. In the case of the hydrogen atom, as the simplest example, the complete wave equation may be written in the form
Im ▽2 x,y,zψ + Iμ ▽2η,μζψ ?8π2μ?ψmh2η2 + μ2 + ζ2
? 4πih?ψ?t = 0, (3)
where x, y, z, are the coördinates of the center of mass of the atom and ξ, η, ζ the coördinates of the electron with respect to the center of mass. If m? and m+ are the masses of electron and proton, m and μ have the significance
m = m? + m+and Iμ = Im? + Im+
. Equation (3) is solved by
ψ = U1(x,y,z) U2(η, ν ζ) ?2πiEth
, where E may have a continuous set of values and represents the total energy. U1 and U2 must satisfy the equations
▽12U1 + 8π2mβU1h2 = 0, (4)
and
▽22U2 + 8π2μh2 (α ? μ?mη2 + ν2 + ζ2)U2 = 0 (5)
, where
α + β + E
.  相似文献   

9.
Capabilities of the resonance flourescence technique are discussed. Specific reference is given to the study of ground state dipole and electric quadrople transitions below about 5 MeV in 206Pb, 207Pb, 208Pb and 209Bi. It is suggested that many of the states observed in 207Pb and 209Bi arise from the weak-coupling of a p12 neutron hole or an h92 proton to collective levels of 208Pb.  相似文献   

10.
End effects on the flow of heat, mass or electrical energy through a cylindrical rod were investigated analytically. Three of the eight analytical solutions were evaluated on a 36091 computer and compared with the results of a numerical analysis package dubbed “Heating III” to show that both compared favorably.Platinum and Hastelloy X rods were subjected to an electrical current applied via two electrodes, one covering one end and the application of the other being thesubject of this investigation. End effects based on voltage deviations of one part in a thousand or more were found to extend for a length to diameter ratio ranging from 0·6 to 1·9 depending upon whether the small electrode position being investigated was at the center of the end or on the periphery of the end. Calculated results from Heating III showed excellent agreement with the experimental results.In addition to their reported applications, the analytical solutions represent a contribution to a neglected area of applied mathematical physics and as such should prove equally useful in other areas that are dynamically analogous. Also the discontinuous infinite integral technique of solution has proven to be a very powerful one.  相似文献   

11.
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12.
Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form
h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)
where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by
λp,T=(v2?v1)(?h?v)P,T= (v2?v1)[(B?TF)+p(D?TH)]
. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.  相似文献   

13.
The essential content of a recent paper by the present writer comprises a comprehensive discussion of the physical bases underlying derivation of formulas for calculating the temperature distribution T, maximum temperature Tm and average temperature Ta in a toroidal electrical coil of rectangular cross section, internally generated heat and change of wire resistance with temperature being taken into account. Illustratively, the solution for the boundary value condition of constant surface temperature and uniform equivalent thermal conductivity was obtained.For the most part, however, problems that arise in practice are not encompassed in the comparatively simple boundary conditions of constant temperature. Experiment shows that in general the boundary condition is T ? T′ = ? K??n; whereof n denotes the outward drawn normal to the coil surface, K = (knh) the ratio of the equivalent thermal conductivity in the direction of n to the emissivity of the boundary surface, and T and T′ are the corresponding temperatures in the coil surface and the immediately adjacent ambient medium. Again, it frequently ensues in practice that the thermal conductivity is substantially different in the directions of the two principal axes of the cross section.In the present paper formulas for T, Tm, and Ta are obtained for electrical coils of ratio of external to internal radius greater than (roughly) two whereof (i) the thermal conductivity is different in the directions of the two principal axes of the cross section, (ii) K is different on but constant over each of the four faces of the coil, and (iii) no restriction is made as to T′ except that over each face it be expressible in a generalized Fourier series. Determination of T is posed as a boundary-value problem in the mathematical theory of heat; the formal solution of T effected by expansions in orthogonal functions; and Tm and Ta then determined through use of their known relationships with T. The resulting formulas are in the form of rapidly-converging singly-infinite trigonometric-hyperbolic series. Illustrative of application of these general formulas, the maximum temperatures in a coil of given dimensions subject to two different sets of surface conditions are calculated and found to be in excellent agreement with the known measured values.The just-mentioned formulas encompass practically all cases encountered in practice except those coils which do not satisfy the restriction as to ratio of radii. For these latter formulas for T, Tm, and Ta are obtained pursuant to conditions of (i) equivalent thermal conductivity different in the directions of the two principal axes of the cross section, (ii) K, and likewise T′, different on but constant over each of the four faces of the coil. These formulas are in the form of rapidly-converging singly-infinite trigonometric- Bessel function (of zero order) series: Illustratively, the maximum temperature in a coil of given dimensions is calculated and found to be in excellent agreement with the known measured value.  相似文献   

14.
The removal of mercury from dilute solutions of mercuric nitrate by mercuric sulfide is indicative of an adsorption process. The relation of log xm vs. log C shows that the Freundlich adsorption isotherm exists.At higher concentrations of mercuric nitrate a reaction takes place with the mercuric sulfide forming a compound of the possible formula 2HgS·Hg(NO3)2 mercuric nitratodisulfide.  相似文献   

15.
If T maps a convex domain DT into itself, and if {ωn} is a real sequence with range in (0, 1] then the recursive averaging process,
Xn+1=(1?omega;n) Xn+ωn+ωnTxn, x0=ξ?DT
generates a sequence {x?n}; with range in DT. Under suitable conditions on DT, T and {ωn} the sequence {x?n} will converge in some sense to a fixed point of T. We prove that if DT is a closed convex subset of a complex Hilbert space H, if Tω = (1 ? ω) I + ωT is a strict contraction for some ω ? (0, 1], and if {ωn} satisfies the conditions,
ωn → 0
and
∑n=0∞ωn=∞
then, for arbitrary ξ ? DT, {x?n} converges strongly to (the unique) fixed point of T. We also prove that if DT and {ωn} satisfy the foregoing conditions, if T has at least one fixed point, and if Tω is non-expansive for some ω ? (0, 1], then for all ξ ? DT, {x?n} converges at least weakly to some fixed point of T. Finally, we apply these results to linear equations involving bounded normal operators and obtain an extension of the classical Neumann operator series.  相似文献   

16.
In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form
d2ydx2+εμ(x)dydx+ω2cy = 0
, where ε is a real parameter less than unity, ωc is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x?2, however, it is shown that the approximate solution is accurate to 0(ε2), as x → 0? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to 0(ε2ln|x||x|) as x → 0?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.  相似文献   

17.
The exact solution of the equation
d2xdt2+dx+d′f(wt)x3=0,
where d, d' and w are positive constants, and ?(wt) is a rectangular periodic function of time is discussed. The equation describes approximately the transversal movement of a particle in an alternating gradient accelerator. The exact solution is obtained in the form of a composite recurrent relation containing five particular solutions. Each of these solutions corresponds to a specific well-defined area of the phase plane of the initial conditions. The dynamical behaviour and the stability of the movement are examined analytically.  相似文献   

18.
The experimental investigation of the laws governing the excitation of x-rays by electron impact appears to be near completion. However, the existence of the proton, with its mass of 1,846 times that of the electron, provides the possibility of introducing anew variable—the mass—into collision experiments. Previous investigations have led to inconclusive results except that it is known that when fast α-particles fall on an element, its characteristic x-ray spectrum is excited.In the writer's experiments, a mass spectrograph is used to bring either protons or electrons, of same energy in the range from 15 kv. to 25 kv., onto a copper target. The arrangement eliminates several known sources of error. X-radiation produced passed into an argon filled Geiger counter.The procedure was to compare two currents: (1) the current composing an electron beam just intense enough to produce detectable x-rays; and (2) that composing the strongest proton beam which certainly produced less radiation.The ratio of these currents is a minimum value for p, the ratio of the efficiencies for excitation of electrons and protons respectively.The results were that (1) no radiation from proton impacts was observed, (2) the minimum value of p from three sets of experiments was 740; 237,000; and 6,400.In view of the directness of the experiment, the order of magnitude of the highest figure is presented as the result, i.e. ? = 105. This may be compared with (M/m = 1,846, W/m)32 = 79,310, (M/m)2 = 3,408,000.Wave mechanical considerations of impact processes indicate that radiation should be produced by proton impacts of energy greater than the critical energy (for copper, 8.86 kv.) but intensity formulas have not been derived.If they can be derived the present results offer the possibility of checking the wave mechanics of impact processes.  相似文献   

19.
The solution of the differential equation y″ + 2Ry′ + n2y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n2 ? R2, it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is (s ? Δn)2π. (b) When n2 ? p2 = R2, there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency Δn2π in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.  相似文献   

20.
The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil
P(s)=s2I+sB+A,
where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy:
(Bx,x)2<4(Ax,x).
A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system,
|tanφ|?[4(Ax,x)?(Bx,x)2]12(Bx,x).
When a comparison between positive operators A and B is available, say B2=KA, then
|tan φ|?√(4?K2)K.
The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.  相似文献   

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