Volume 17, Issue 12, 01 December 1946

Report of the Electron Microscope Society of America's Committee on Resolution
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Electron Microscope Examination of the Microphysical Properties of the Polymer Cuprene
View Description Hide DescriptionThe microphysical structure of cuprene is described from electron micrographs and some suggestions are made concerning the mechanism of the formation of this substance. Cuprene samples prepared by polymerization of acetylene in the presence of finely divided cuprous oxide are shown to be composed of hollow fibers characterized by both longitudinal and transverse structure. The striking appearance in electron micrographs of this type of cuprene serves to identify it in clogging deposits taken from acetylene lines. This is illustrated by a deposit taken from a reactor used in the process of hydrogenation of acetylene to ethylene. Cuprene specimens formed by alpha‐ray bombardment of acetylene gas are shown to be composed of round particles joined by short, straight, thick necks which are about 500 to 1000A in length. These are samples of cuprene which have been bombarded further by alpha‐particles after their initial formation. An oxidized sample has round particles of mean diameter 4900A; an unoxidized sample has a mean diameter of 3800A for the round particles. There is evidence to suggest that alpha‐ray cuprene is liquid in nature. The two types of cuprene are so different physically that a change in nomenclature is in order.

A Machine for the Application of Sand in Making Fourier Projections of Crystal Structures
View Description Hide DescriptionThis paper presents a method of using sand in building models of two‐dimensional Fourier projections in crystal structure analysis. The machine for producing these models is described and the results are illustrated by projections of known structures. The mathematical principle of this method of summing Fourier series is identical to that which Bragg developed in England. The chief difference is in the use of sand instead of photographic film as the medium for producing contrasts in electronic density throughout the projection. The method herein described has a number of features which contribute to its usefulness: A two‐dimensional Fourier projection comprising the addition of one hundred and fifty terms may be made in about four hours by a single worker. The time required is not determined by the size of the unit cell. The subdivisions of the projection face are sufficiently fine to assure an accuracy as great as the x‐ray data from copper radiation will warrant when a scale of one angstrom to two centimeters is used. Upon embarking on a new structure problem, no changes in the equipment are necessary; as the initial set of templates suffices for all future work. The minimum interplanar spacing that the machine will accommodate is determined by the characteristics of copper radiation rather than the order (h, k, or l) by which the spacings are characterized. The resulting sand model of the projection is the same shape as the unit cell projection with regard to both the angle between the edges and the ratios of the edges of the cell. When the x‐ray data warrant, a distinction can be made between heavy atoms and lighter atoms by the differences in the height of the sand. Photographs of the projections (see Figs. 7 and 9) portray atoms as distinct spots.

Radar Reflections from Long Conductors
View Description Hide DescriptionThe distribution of the backscattered radiation of a long straight conductor shows a sharp return lobe perpendicular to its orientation. In horizontal observation such a ``rope'' would therefore give an appreciable echo only if it were almost exactly vertical. In practice no such effect will be observed since, because of wind and other disturbances, the rope will never be straight but actually over its whole length will pass through regions of space where the phase of the incident wave will change many times through its full range 2π. The effect of these irregularities will be to make the sharp dependence of the cross section upon the orientation of the rope disappear, and to give a cross section proportional to the total length L of the rope. In Section I, the wave‐length λ is assumed to be large compared to the transverse dimensions of the rope. In Section II, the results are extended to include the case where the wave‐length λ of the incident radiation is not necessarily large but is comparable with or even small compared to the width d of a ribbon‐shaped rope. It is found that the cross section of such a rope can be written generally in the form σ = KAf(πd/λ), where K is a numerical constant depending upon the deviations of the rope from the vertical and where A = Ld is the area of the ribbon. The length L is assumed to be much greater than λ. The function f is evaluated by the use of a method developed by Morse and Rubenstein.

Viscometric Investigation of Dimethylsiloxane Polymers
View Description Hide DescriptionThe intrinsic viscosities of an extended series of linear methyl polysiloxane fluids, having dimethylsiloxane (Me_{2}SiO) as the repeating unit, have been determined in toluene solution. These values have been correlated with their molecular weights, determined chemically and by osmotic pressure and light scatteringmeasurements. The Staudinger equation was found applicable only for fluids of relatively low molecular weights, i.e., up to about 2500; for the higher polymers, the best correlation of intrinsic viscosity with number average molecular weight was found to be:.This held reasonably well for molecular weights from 2500 to about 200,000. The bulk viscosities of the polysiloxane fluids were found to conform to the Flory relationship for melt viscosities; the expression:appears to be reasonably valid for molecular weights above 2500.

Theory of Small Signal Bunching in a Parallel Electron Beam of Rectangular Cross Section
View Description Hide DescriptionThe phenomenon of bunching which takes place in a velocity modulated electron beam can be described in purely kinematical terms if space charge effects are negligible. However, the non‐uniform distribution of charge in the bunched beam gives rise to a field which opposes the bunching process so that the kinematicsolution may be validly applied only for a limited length of drift space. An accurate solution of the bunching process requires the integration of the dynamical and field equations. These reduce to a linear homogeneous system under the assumption of ``small signal'' conditions. The device of a high frequency ``surface charge'' is employed in formulating the boundary conditions at the surface of the beam. In the non‐relativistic approximation the dynamical and field equations yield solutions which are classified as non‐solenoidal or solenoidal accordingly as the motion produces or does not produce a high frequency charge density within the beam. There exists a non‐solenoidal type of motion which generates no field outside the beam. The actual physical problem is solved by taking a suitable linear combination of both solutions. Under conditions met with in practice, a large part of the actual solution may be of the solenoidal type. The high frequency component of beam current appears as a sum of a volume current and a ``surface current,'' the latter term arising from the longitudinal motion of the high frequency ``surface charge.'' Again, under conditions met with in practice, a large part of the high frequency component of beam current may occur in the form of ``surface current.'' The theory contains three debunching wave numbers: one identical with that which occurs in Webster's debunching theory for a beam of unlimited cross section, and two which depend upon the transverse dimensions of the beam and drift tube, the current density, frequency, and beam velocity.

On Vibrations of Shallow Spherical Shells
View Description Hide DescriptionThe problem of the axi‐symmetrical vibrations of a shallow spherical shell is reduced to two simultaneous differential equations for the tangential and normal components of displacement. The solution of these equations is obtained in terms of Bessel functions. With this solution the third‐order determinant for the frequencies of a shell segment with clamped edge is given but not evaluated. Instead, an approximate value for the lowest frequency is calculated by means of the Rayleigh‐Ritz procedure [Eq. (40)]. It is found that very little curvature is needed to modify appreciably the corresponding flat‐plate frequency. The approximations which are made are those customary for shallow shells: (1) omission of the transverse shear term in the tangential force equilibrium equation, and (2) relations between couples and changes of curvatures as in the theory of flat plates.

Non‐Uniform Transmission Lines and Reflection Coefficients
View Description Hide DescriptionA first‐order differential equation for the voltage reflection coefficient of a non‐uniform line is obtained and it is shown how this equation may be used to calculate the resonant wave‐lengths of tapered lines.

Resonant Frequencies of the Nosed‐In Cavity
View Description Hide DescriptionIn this paper the resonant frequency of the nosed‐in cavity is studied as a function of the cavity dimensions. Maxwell'sdifferential equations and boundary conditions are converted into an integral equation which is solved approximately by the Ritz variational method. The size and shape of the cavity are fixed by specification of the dimensions (cf. Fig. 1) r _{1} and r _{2}, the inner and outer radii; ε^{I}, the post length and ε^{II}, the gap space. If the cavity is to resonate to the wave‐length λ, only three of its dimensions can be given independently; the fourth will be a function of the given three and the wave number k=2π/λ. For fixed r _{1}/r _{2} the dependence of kε^{I} on kε^{II} is calculated with a precision of 1 percent.

The Study of a Certain Type of Resonant Cavity and Its Application to a Charged Particle Accelerator
View Description Hide DescriptionA linear accelerator for charged particles, consisting of a number of cavity resonators placed end to end with the inside end plates removed, and thus forming a wave guide with two closed ends, is discussed. The shape of each section must be so chosen that the velocity of the particle will correspond to the phase velocity of the radiation of the cavity. In order to determine possible shapes for the cavity, the stationary TM _{01} modes between two parallel conducting planes are determined. In Part 1, excitation of a single mode is discussed, including the shape of the conducting surface that can be used to connect the planes and thus form a cavity. Three cases are considered—when the phase velocity in the cavity is greater than, equal to, and less than that of light. If only one mode is excited, and the phase velocity is not greater than that of light, then the radius of the cavity becomes infinite at the two plates. In Part 2, the simultaneous excitation of two modes is considered. The phase velocity of one mode is that of light; the other has a phase velocity 1/n that of light, where n is an odd integer. The cavity can be made finite by a suitable choice of the relative amplitude of the two modes, and the radius of one end of the cavity. This requires a rather elaborate discussion. This case is of particular interest for the acceleration of electrons. It is shown, furthermore, that only the first mode contributes to the acceleration of the electron. In Section 4, the application of the methods used in Part 3 to the case of an accelerator for heavy particles is touched upon.

A Millimeter‐Wave Reflex Oscillator
View Description Hide DescriptionThe factors which affect the frequency limitation of reflex oscillators are discussed. General design considerations are given for the construction of oscillators of the shortest possible wave‐length. The performance of several tubes is given. The highest frequency obtained was 72,000 megacycles.

Resonant Circuit Modulator for Broad Band Acoustic Measurements
View Description Hide DescriptionA modulation method is described whereby a broad band frequency response is obtained for recording of sound. In particular low frequency sound approaching 0 c.p.s. can be recorded. The theory of the resonant circuit modulating principle is first discussed followed by a description of the apparatus which was constructed for this purpose.

Transient Temperatures around Heating Pipes Maintained at Constant Temperature
View Description Hide DescriptionThis paper deals with the transient temperature distribution caused by conduction around cylindrical heating pipes maintained at definite constant temperatures. The theoretical equation for this case known from the literature is integrated numerically and presented in graphs. The amounts of heat dissipated are also computed and presented. As an application, the influence of pipe diameter upon temperatures obtained at given times is shown for a particular case. For practical installations, the characteristics of which usually deviate from those upon which the theoretical equation is based, the graphs may be used for obtaining close estimates as an aid in designing the installation. It is shown that on two actual installations a satisfactory agreement between measured and computed temperatures (maximum deviation about 10 degrees F) was obtained.

The Effect of Temperature on the Strength and Fatigue of Glass Rods
View Description Hide DescriptionBy the use of an electrodynamic quick‐loading device, the strength of scratched soda‐lime glass rods was measured at various temperatures between − 190°C and 520°C for load durations of 0.1 second, 10 seconds, and 100 seconds. It was found that the strength was highest at the lowest temperature, declined to a minimum at 100°–200°C, and then increased for higher temperatures. Static fatigue was exhibited at all temperatures, being maximum in the region of minimum strength. Comparisons with the results of other experimenters are made.

Plastic Flow, Creep, and Stress Relaxation: Part I. Plastic Flow
View Description Hide DescriptionPlastic substances are considered to be composed of units of flow with various yield values. It is shown that in this case the product of the strain rate and viscosity is equal to the sum of the differences between the applied stress and the yield values. This relationship can be applied to any plastic system free of elastic after‐effect and expresses their mechanical properties in terms of a coefficient of viscosity which is independent of the stress applied. With the proper choice of the distribution of yield values any kind of relation between stress and strain rate can be established. This relationship is applied to plasticflow which is defined as a deformation mechanism having a curvilinear relationship between stress and rate of deformation and a constant rate of deformation at constant stress. Equations are given for the coefficient of viscosity of such systems and for the relaxation of stress at constant deformation as a function of time.

Plastic Flow, Creep, and Stress Relaxation: Part II. Creep
View Description Hide DescriptionThe general equation covering the deformation of plastic substances given in Part I is applied to creep.Creep is defined as a mechanism of deformation for systems which have a curvilinear relationship between stress and strain rate and a curvilinear relationship between strain and time at constant stress. Creep is connected with changes in the internal structure of a plastic substance and results in an increase in strength of such materials through work‐hardening. Equations are derived which give the stress as a function of strain rate and time (time‐hardening), as a function of strain rate and strain (strain‐hardening) and as a function of strain rate, strain, and time. The difference between time‐hardening and strain‐hardening is discussed. Expressions are given for the coefficients of viscosity of such systems which are independent of the stress applied. The relaxation of stress at constant strain is discussed, and it is shown that the stress relaxation depends upon the history of the substance under test. The concept of creep is also applied to thixotropic systems which are considered as cases of work‐softening.

Plastic Flow, Creep, and Stress Relaxation: Part III. Creep and Elastic After‐Effect
View Description Hide DescriptionA large number of substances show the phenomenon of elastic after‐effect, and part of their deformation recovers on unloading as a function of time. This portion of the deformation at constant stress has a strain rate which decreases with time and is therefore comparable to creep. Expressions are given for the strain‐time relationships of such systems, and the process of stress relaxation at constant strain is discussed. The equations given in connection with plasticflow,creep due to work‐hardening, thixotropy, and creep in combination with elastic after‐effect are applied to data given in the literature, and it is shown that these equations suffice to describe the deformation and relaxation mechanisms of a variety of materials such as metals, clay soil, food products, acrylic acid polymeride, polyvynil chloride, cellulose acetate, manila ropes, paper laminates, phenolic molding compounds, rubber, asphalt, and bituminous pavements.

Asymptotic Solutions for the Normal Modes in the Theory of Microwave Propagation
View Description Hide DescriptionIn this paper are presented several extensions of the ``W.K.B.'' method in the asymptotic solution of differential equations. The developments were made in the course of a systematic study of the application of asymptotic solutions to the normal‐mode theory of microwave propagation. In Section 1 a brief outline is given of the now standard normal‐mode theory of propagation of microwaves in an atmosphere with a horizontally stratified index of refraction. Particular emphasis is given in this study to the case of a ``surface duct.'' The case of ``leaky modes'' is studied in Section 2. Here the aim has been to obtain explicitly the terms after the leading one in the asymptotic expansion of the solution, in order to have an estimate of the order of magnitude of the error introduced by the use of the leading term only. This is achieved in Eq. (20), which is of general applicability to a wide class of physical problems. The first correction term to the phase‐integral solution for the characteristic values of the normal modes is set out in Eqs. (28) to (30). An alternative asymptotic solution for the case of leaky modes, which includes the first correction terms, is given in Eqs. (30), (40), and (41). In Section 3 a direct method is given of determining the characteristic values from the phase‐integral relation. If the modified index of refractiony(h) is given by a power series (49) then with the aid of Eqs. (50)–(52) one can compute directly the characteristic value Λ_{ m } by Eq. (46), using (47) and (48). The degree of success of this method is illustrated in Table I. The principal contribution of this study is in the development of asymptotic solutions for the case of transitional modes which are on the border line between the leaky modes and the trapped modes. The standard ``W.K.B.'' method is inapplicable in this case. The results for the determination of the characteristic values and height‐gain functions are as follows. One first obtains ν_{0} defined in (85), which in the case of an exponential model can be facilitated by the use of Fig. 2. The quantities γ_{ m } ^{0}, dγ_{ m } ^{0}/dν_{0}, and [(∂P(ν_{0})/∂ν) . (∂P(ν_{0}) / ∂γ_{ m } ^{0})] are then read off Table IV. The characteristic value is then determined from (87), (88), and (68), particular examples of which are given in (89) and (90) for the ``exponential model,'' and in (94) and (96) for the ``power law'' model. These include a correction term Δ_{ m } which was obtained from the perturbation theory of transitional modes developed in Section 4D. A comparison between exact characteristic values and those obtained from the theory of transitional modes is shown in Tables II and III for the ``exponential'' and ``power law'' models, respectively. It is seen that the theory of transitional modes developed here yields accurate characteristic values over a wide region, extending into the leaky modes on one side and into the trapped modes on the other side. The height‐gain functions can be obtained from (60), (69), (76), (77), (83), and (101). In the case of an exponential model the variables u and v can be computed with the aid of Fig. 2. The asymptotic theory of trapped modes developed in Section 5 follows Langer's method, whereby the difficulty with the Stokes phenomenon is completely obviated. The procedure used is to develop the solutions around the two turning points h _{1} and h _{2} shown in Fig. 1 and to join the solutions at the duct height h _{0}. The relation determining the characteristic values thus obtained is (140), which bears a similarity to the corresponding equation for a ``bilinear'' model. Indeed, Eqs. (131), (132), (136), and (137) give the transformation required to turn the model with a continuous y(h) curve shown in Fig. 1 into one where y(h) is represented by two straight lines that intersect at h _{0}. It is shown in Eq. (159) how one can derive the Furry‐Gamow formula for the decrement of completely trapped modes from Eq. (140). Equations (160) to (163) give expressions for the height‐gain functions for highly trapped modes. These cover the whole range of height including the vicinity of the two turning points.
 LETTERS TO THE EDITOR


Concerning ``Computer for Solving Linear Simultaneous Equations''
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Preparing Pigments for the Electron Microscope
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