US 3765773 A Abstract Systems and components are disclosed for the utilization of electromagnetic waves whose phase difference, between lattice planes in discrete phase-ordered media, is appreciable in the forward-scattered direction.
Claims available in Description (OCR text may contain errors) United States Patent [191 Weiner Oct. 16, 1973 SYSTEMS AND COMPONENTS FOR THE [56] References Cited UTILIZATION OF ELECTROMAGNETIC UNITED STATES PATENTS WAVES IN DISCRETE PHASE'ORDERED 3,563,630 2/1971 Anderson et al 350/96 wo MEDIA [76] Inventor: Melvin Milton Weiner, 56 Marcellus Primary ExaminerRonald L. Wibert Dr., Newton Centre, Mass. 43614 Assistant Examiner-Conrad Clark [22] Filed: 0c. 5, 1970 Attorney-Charles Hleken Systems and components are disclosed for the utiliza- [52] Cl -r 356/114 356/111 350/96 tion of electromagnetic waves whose phase difference, [51] Int. Cl. G01n 21/40, GOlh 9/02 between lattice planes in discrete phase ordered [58] F leld of Search 350/96; 335s66/llllo46, dia is appreciable in the forwardscattered direction 8 Claims, 30 Drawing Figures PATENTEU UN 16 I973 3.765773 SHEET 1!]? 9 FIG. ICL FIG. lb Flg c Pml m nnm 15 mars 3.765773 SHEET 5 OF 9 I I 1 m LY b,o TOTAL REFLECT. m Y PARTIAL TRANS. II: I I 1.0 IS,/T,I2A M UM. FIG. /2 ISJTIZ ,1; I i AM Pmmeunm 16 ms 3.765773 SHEET 7 OF 9 FIG. /5 SYSTEMS AND COMPONENTS FOR THE UTILIZATION OF ELECTROMAGNETIC WAVES IN DISCRETE, PHASE-ORDERED MEDIA BACKGROUND OF THE INVENTION This invention relates to the utilization of the transmission, reflection, dispersion, and polarization properties of electromagnetic waves, over the entire electromagnetic spectrum, in discrete, phase-ordered media. The theory of electromagnetic waves in discrete media is disclosed by the applicant in Appendix A. The concept and definition of phase-ordered media are also disclosed in Appendix A. Phase-ordered media include continuous media, single crystals, liquid crystals, polycrystalline and amorphous media in which the phase differences between scattering centers are small, and artificial lattices consisting of media in which obstacles are periodically imbedded or media which are periodically modulated by an elastic wave or other incident wave. Specific properties of interest in this invention fall into one of two categories. The first category includes the transmission bands, reflection bands, frequency dispersion, and birefringence which occur at sufficiently small grazing angles of incidence in discrete, phase-ordered dielectric media. The second category includes the increased skin depth and reduced conduction loss of electromagnetic waves at arbitrary angles of incidence in discrete, phase-ordered metallic media. ln particular, this invention relates to metallic radiation guides, such as wires, transmission lines, waveguides, and cavities, in which travelling or standing electromagnetic waves propagate in well-defined modes with minimum radiation loss. This invention also relates to dielectric radiation guides, such as fibers and thin films, in which electromagnetic waves propagate by successive multiple reflection with minimum radiation loss. This invention also relates to systems and components for cont'rol, communication, display, dataprocessing, information storage, and detection in which precise filtering, switching, or modulation of the direction, wavelength, amplitude, or polarization of the incident electromagnetic radiation is desired. This invention further relates to.-systems for the investigation of lattice structure and lattice defects of materials. Still further, this invention relates to the detection of externally applied energy which modifies the lattice structure or the electrical properties of a discrete, phaseordered medium. The present state-of-the-art of most electromagnetic systems and components is characterized by wave interactions in which the phase difference between scattering centers of the interacting medium can be neglected or assumed to be zero.- The medium is therefore usually referred to as being continuous." The wave interactions for continuous media can differ significantly from the wave interactions for discrete phaseordered media in which the phase difference between scattering centers is appreciable. A detailed comparison, of the properties of electromagnetic waves in continuous and discrete phase-ordered media, is given in Appendix A. This invention relates to a specific class of wave interactions in discrete phase-ordered. media. This invention specifically relates to waves whose phase difference, between lattice planes in discrete phase-ordered media, is appreciable in the forward direction. (see Appendix A, Conditions V 1, s IET This invention should not be confusd with wave interactions in the present state-of-the-art of periodic media. One such class of wave-interactions is characterized by Bragg reflection in which the phase difference between lattice planes isappreciable in the reflected direction (see Appendix A,). Another class of wave interactions in the present stateof-the-art of periodic media is characterized by wavelengths which are large compared to the spacings between scattering centers of an artificial lattice (see W. E. Kock, Metallic Delay Lenses Bell Syst. Techn. J. 27, 58, 1948). A further class of wave interactions in the present state-of-the-art of periodic media is restricted to waves which are incident at normal incidence on artificial dielectrics (see S. B. Cohn, J. App. Phys. 20, 257, I949; 21, 674, 1950; 22, 628, 1951; see also H. S. Bennett, The Electromagnetic Transmission Characteristics of the Two- Dimensional Lattice Medium, J. App. Phys. 24, 785, 1953), In this latter case of wave interactions, the phase difference between lattice planes can be appreciable in the forward direction. A-lso related to this latter class of wave interactions is the propagation of waves in periodic-loaded transmission lines in which the incident and exit directions of propagation are known (as in the case of'normal incidence). (See L. Brillouin, Wave Propagation in Periodic Structures, New York: Dover, 1953, Chapters 9, l0, and Appendix). This invention relates to two classes of wave interactions in which the phasedifferences between lattice planes is appreciable in the forward direction and which have not been utilized in the present state-of-theart of electromagnetic systems and components. The first class of wave interactions related to this invention consists of electromagnetic waves incident at small grazing angles of incidence on a discrete phase-ordered dielectric. The second class of wave' interactions related to this invention consists of electromagnetic waves incident at an arbitrary angle of incidence on a discrete phase-ordered metal. The unique-electromagnetic properties related to this invention for the first class of wave interactions,in phase-ordered dielectrics,are: a. The effective phase velocity of wave propagation is a function of the angle of incidence, polarization, and incident wavelength for a given dielectric constant and lattice spacing of the medium. b. Narrow transmission bands occur for angles of incidence sufficiently small. c. Total reflection occurs in relatively wide bands, for angles of incidence sufficiently small, regardless of whether the relative phase velocity is less than or greater than unity. - The unique electromagnetic properties related to this invention for the second class of wave interactions, in phase-ordered metals, are: a. The effective phase velocity of wave propagation is weakly dependent upon the DC conductivity and is strongly dependent upon the lattice spacing for media of large DC conductivity. b. The effective skin depth is larger (and the effective surface resistivity is smaller) than for continuous metals. One component of this invention, common to all claims and embodiments, is a phase-ordered medium, whether it be a dielectric or a metal. A phase-ordered medium is one which satisfys the conditions of Equation 2, 7 of Appendix A and includes .those media which have been identified above. If the incident electromagnetic radiation penetrates to only a limited effective distance within the medium, then it is understood that the conditions of Equation 2, 7 need only be satisfied by that portion of the medium to which the incident radiation penetrates. Therefore a medium with only short range order may be phase-ordered. This is particularly applicable for radiation incident at small grazing angles of incidence because an appreciable percentage of the incident radiation may be reflected by a single lattice plane. A second example of a phaseordered medium which may have only short range order is radiation incident on metals with a skin depth of penetration comparable to a few lattice spacings. The scattering centers of a phase-ordered medium j may consist of either a. atomic or sub-atomic particles, such as electrons, atoms, ions, molecules, unit cells of a crystal, which satisfy the conditions of Equation 2,7 of Appendix A; b. obstacles which are periodically imbedded in a dielectric to form an artificial lattice; c. atomic or sub-atomic particles of a medium which is periodically and spatially modulated by an elastic wave or other wave propagating in the medium, such as an acoustical wave which spatially modulates a liquid crystal. A phase-ordered medium j mayexist as a solid, liquid, gas, or plasma provided that the conditions of Equation 2.7 of Appendix A are satisfied. Although the applicant defines and introduces in Appendix A the concept of phase-ordered media, this application does not claim the invention ofphase-ordered media nor does it claim the invention of new forms of phaseordered media. Rather, this application claims new and novel ways of utilizing the properties, disclosed by the applicant in Appendix A, of electromagnetic waves in phase-ordered media. In defining a phase-ordered dielectric and a phaseordered metal, it is convenient to refer to the paramet wr-wWr-(m we) relative wave phase velocity for continuous media where Y v; wave phase velocity in medium i if medium i were continuous v, wave phase velocity in were continuous m, y. permeabilities of the respective media if the respective media were continuous q, s, permittivities of the respective media ifthe respective media were continuous. The parameter v is generally a complex number since most media have some absorption loss. if Re v lm v (the real part of v is much larger than the imaginary part of v then medium j shall be referred to as a dielectric with respect to medium i. If Im v Re v then medium j shall be referred to as a metal with respect to medium 1'. A phase-ordered dielectric is therefore, for purposes of this invention, a medium j which satisfys the conditions of Equation 2.7 of AppendixA and for which (lm v /Re v 1. A phase-ordered metal, for purposes of this invention, is a medium j which satisfys the conditions of Equation 2.7 of Appendix A and for which (lm v /Re v 1. Total reflection or total" transmission by a dielectric shall be interpreted as referring to anidealized dielectric with medium j if medium j BRIEF SUMMARY OF THE INVENTION According to the invention, there is apparatus for selectively propagating waves in a direction having a component parallel to an interface comprising means defining discrete phase-ordered media along the interface characterized by a wave refraction constant, such as the dielectric constant, permittivity, permeability or index of refraction, and a spacing d between lattice planes generally parallel to the interface, a source of waves, and means for establishing the wavelength, polarization and angle of incidence upon the inter-face of the waves to selectively propagate the waves in a direction having a component parallel to the interface in a predetermined manner controlled by the wave refraction constant and the spacing d and establish a substantial phase difference between wave fronts at adjacent ones of the lattice planes in the forward scattered direction at least of the order of a radian. One embodiment of the invention is a metallic radiation guide, for the propagation or containment of electromagnetic waves in well-defined modes, whose conductors and boundary walls comprise a discrete phaseordered metal. The geometry and configuration of the radiation guide may comprise that of any existing wire, transmisssion line, waveguide, cavity, or electrical circuit which guides electromagnetic radiation by means of one or more metallic conductors. The claim of this invention is that the metallic conductors (media of large DC conductivity) of the radiation guide are discrete and phase-ordered. A preferred embodiment of this invention is that the phase-ordered metals have maximum DC conductivity with maximum spacing of the lattice planes parallel to the guide wall. Such a preferred embodiment has minimum surface resistivity in metallic radiation guides of well-defined modes of propagation (see Equation 4.67 and Table 7 of Appendix A). Metallic conductors, of present radiation guides of well-defined modes of propagation, are usually electroformed or electroplated metals consisting of randomly oriented crystals with an effective lattice structure of large phase disorder. Such metallic conductors are usually treatedin practice as if they were continuous metals with zero phase difference between scattering centers. Table 7 of Appendix A demonstrates that the surface resistivity for radiation guides of this invention, which utilize discrete, phase-ordered metals, is less than for those which utilize continuous metals. This invention is applicable to metallic radiation guides in which the radiation is guided in well-defined modes and is not applicable to metallic-radiation pipes in which the radiation propagates by multiple reflection in undefined modes. The cross-section of the radiation guide and of the phase-ordered metal may be of any geometry, including rectangular and circular geometries, appropriate'for a given mode of propagation. The phaseordered metal should have a thickness at least equal to the effective skin depth given by Equation 4.66 of Appendix A. The discrete, phase-ordered metal may adjoin or be clad by dielectric media which are internal or external to the radiation guide and which are not necessarily phase-ordered. The discrete, phase-ordered metal may also adjoin or be mechanically supported by a metallic medium which is external to the radiation guide and which is not necessarily phase-ordered. The metallic radiation guide of this invention is particularly well-suited to microminiaturized, high frequency electric circuits such as microcircuits and integrated semiconductor circuitry because: a. only small amounts of phase-ordered metal are required; b. minimum surface resistivity is required because of 5 the small size of the conductors; and c. the phase-ordered metal is compatable and may be integrated with the single-crystalline media associated with microminiaturized circuitry. A second embodiment of the invention is a dielectric radiation pipe, for the propagation of electromagnetic waves by successive multiple reflection within certain defined bands of grazing angles of incidence, whose boundary walls comprise a discrete, phase-ordered dielectric. If til is the grazing angle of incidence, then the defined bands of grazing angles of incidence are #1 s 4: s 111 where i11 and w are given by Equation 4.43s of Appendix A and are the angles corresponding to the boundaries of the total reflection regions for electromagnetic waves incident on discrete, phaseordered dielectrics. The most common form of the present state-of-the-art of dielectric radiation pipes comprises a dielectric fiber or clad dielectric fiber through which radiation propagates by multiple reflection at grazing angles of incidence less than the critical angle 111,; of total reflection. In the present state-of-theart, the cladding medium j is assumed to have the properties of a continuous dielectric whose index of refraction n, is less than the index of refraction n, of the fiber i. The claims of this invention are that the cladding medium j is a discrete, phase-ordered dielectric and that the grazing angles of incidence are restricted to the total reflection bands of the discrete, phase-ordered dielectric. The index of refraction, of the cladding medium j of this invention, may be greater than the medium i about which it is clad. A preferred embodiment of this invention is that the medium i, through which the radiation propagates, is free space and that the cladding medium j is a discrete, phase-ordered dielectric whose walls serve as the boundary of the free-space medium 1'. The cross-section of the radiation guide may be of any geometry, including tubular and slab geometries, in which the radiation can propagate by successive multiple reflection at constant angles of incldence. The discrete, phase-ordered medium j should be of sufficient thickness, as determined by Section 5.0 of Appendix A, so that total or almost total reflection is achieved. The discrete, phase-ordered medium j may be mechanically supported by and adjoin a dielectric or metallic medium which is external to the walls of the radiation guide. The dielectric radiation guide of this invention is particularly well-suited to present applications which utilize thin films and fibers as radiation guides. The conditions for total reflection at the surface of a discrete, phase-ordered dielectric may be satisfied for sufficiently small grazing angles of incidence, although it should be clear from Equation 4.430 of Appendix A that not all small grazing angles of incidence satisfy the conditions for total reflection. The advantages of this invention over the present state-of-the-art are a. that the cladding medium j may be of larger index of refraction than the fiber medium i; b. the radiation is reflected more completely at the surface of the cladding medium j for small grazing angles of incidence; c. the absorption loss of the fiber medium i can be eliminated completely since the medium i can be free space. It should be understood that the index of refraction, of either the fiber medium i or the cladding medium j, may be varied by electrical or mechanical means for the purpose of filtering, switching, or modulating the incident radiation. It should be further understood that the conditions for total reflection at the surface of a discrete phase-ordered dielectric are a function of the direction, wavelength, and polarization of the incident radiation. Therefore, the dielectric radiation guide of this invention can be used to filter the direction, wavelength, or polarization of the incident radiation. It should be still further understood that several dielectric radiation guides of this invention may be arranged in an array or bundle for purposes of control, communication, display, data-processing, information storage, or detection of the incident radiation. For such a case, a preferred embodiment of this invention is a swiss cheese configuration in which the cheese" is the dis crete, phase-ordered dielectric j common to all of the radiation guides and the holes in the cheese are the media i through which the radiation propagates. A third embodiment of the invention is an electromagnetic band pass filter, comprising a phase-ordered dielectric material, which transmits very narrow bands of radiation of a given wavelength and polarization at specific small grazing angles of incidence. The angles of incidence 111 (and the corresponding wavelengths and polarization), at which the filter is totally transmitting, areil1=|l1 M=i 1,:2, ,where ili isgiven by Equation 4.42b of Appendix A. The 3 db transmission bandwidths Alllm associated with the angles 111 of total transmission, are given, for normal polarization, by Equation 4.46 of Appendix A. Since the angle i11 is a function of the polarization, the band pass filter may also serve as a polarizer. In the present state-ofthe-art, total transmission in a dielectric medium is achieved either for the relative index of refraction v l or for parallel polarization at Brewsters angle Brew," (1%,, p cot v (see Equation 4.42a of Appendix A). Brewsters angle (41 is usually a relatively large grazing angle of incidence. The claim of this invention is that the dielectric medium is phaseordered at small grazing angles of incidence and in particular at the angles I11 so that the dielectric behaves as a narrow band pass filter. A preferred embodiment of this invention is a discrete, phase-ordered dielectric upon which radiation is incident from free space and from which the radiation, within the pass band of the filter, is transmitted into free space where it may be detected or further processed according to the particular application. Another preferred embodiment of this invention is a discrete phase-ordered dielectric which is adjoined by an absorbing detector at theexit surface of the dielectric. For both of these embodiments, the thickness of the phase-ordered dielectric does not appreciably affect the transmission properties of the dielectric. The phase-ordered dielectric may be of arbitrary geometry. However, a rectangular slab or disc geometry is preferred. This invention is particularly suited to applications requiring thin films and miniaturized circuitry. It should be understood that the filter may be one component of a larger system of components for purposes of control, communication, display, data-processing, information storage, or detection of the incident radiation. It should be further understood that the index of refraction or the effective lattice spacing of the phase-ordered dielectric may be designed to be variable with the application of an externally applied electrical voltage or mechanical stress. If such is tha case, then the filter acts as a switch or modulator of the transmitted radiation. A further embodiment of the invention is an electromagnetic system, for the investigation of the lattice structure and lattice defects of materials, comprising a source of electromagnetic radiation, a discrete, phaseordered dielectric whose lattice structure is to be investigated and upon which the radiation is incident at small grazing angles of incidence,'and a detector of the narrow band or bands of radiation which are transmitted by the phase-ordered dielectric. In this embodiment of the invention, the phase-ordered dielectric, whose lattice structure is being investigated, behaves as the filter of the preceding embodiment. Therefore the transmission bandwidths are Aim, at incident grazing angles #1 of total transmission where M i1, :2, The effective lattice spacing, of the lattice planes parallel to the interface, is determined from Equation 4.421) of Appendix A, assuming that-the incident wavelength, angle of incidence, and the relative index of refraction are determined independently. The reduced transmission, which is detected at an expected angle :11, of total transmission, is an indication of the lattice defects or phase disorder of the dielectric. In the present state of the art, the investigation of lattice structure is usually achieved by either 'x-ray Bragg reflection, electron diffraction, or neutron diffraction. The latter two methods utilize particle bombardment of the specimen whereas the first method utilizes electromagnetic radiation of the specimen at sufficiently large angles of incidence 111 (see Equation 4.740 of Appendix A) so that the phase difference between lattice planes is approximately 21r radians in the reflected direction. The claim of this invention is that electromagnetic radiation, which is not necessarily restricted to x-ray wavelengths, is incident on the specimen at sufficiently small grazing angles of incidence 11: so that the phase difference between lattice planes is approximately 21r radians in the forward scattered direction. A preferred embodiment of the invention comprises a collimated, monochromatic source of optical radiation, a slab of the phase-ordered dielectric specimen, andan optical detector of the radiation transmitted by the specimen. The wavelength or the angle of incidence may be varied through a calibrated range of values. Embellishments such as Michelson interferometry may also be employed. A fifth embodiment of the invention is an electromagnetic mirror, comprising a discrete phase-ordered dielectric material, which reflects radiation for specific bands of small grazing angles of incidence. The specific bands of grazing angles'of incidence are 41, 1!; , s [p which are defined in the description of the second embodiment of this invention. The electromagnetic mirror of this fifth embodiment behaves in the same manner as the boundary walls of the dielectric radiation pipe of the second embodiment of the invention. In the present state-of-the-art of dielectric mirrors, the dielectric material is usually either a multilayer of dielectrics whose thicknesses are critical with respect to the incident wavelength or is a dielectric which is assumed to be continuous with suitably small index of refraction so that total reflection can occur at grazing angles of incidence less than the critical angle of total reflection. The claims of this invention are that the dielectric is phase-ordered and that the grazing angles of incidence are restricted to the total reflection bands 111 5 41 5 111 of the discrete phase-ordered dielectric. The index of refraction of the phase-ordered dielectric may be greater or smaller than that of the medium from which the radiation is incident. A preferred embodiment of this invention is that the radiation is incident from free space on a phase-ordered dielectric which may adjoin, for purposes of mechanical support, a dielectric or metallic medium. The surface of the mirror may be planar or curved depending upon whether the focal point of the mirror is infinite or finite. The phase-ordered dielectric should be of sufficient thickness, as determined by Section 5.0 of Appendix A, so that total or almost total reflection is achieved. The relative index of the refraction of the phase-ordered dielectric may be varied by electrical or mechanical means for the purpose of filtering, switching, or modulating the incident radiation. Since the reflection bands of the mirror is a function of the direction, wavelength, and polarization of the incident radiation, the mirror of this invention may be used as a selective filter of the direction, wavelength, or polarization of the incident radiation. A sixth embodiment of the invention is an electromagnetic transducer of externally applied stress, comprising a source of electromagnetic radiation, a discrete, phase-ordered dielectric upon which the radiation is incident at the narrow pass bands of the dielectric at sufficiently small grazing angles of incidence, detectors of the radiation transmitted and reflected by the dielectric in the forward andreflected directions respectively, and efficient means of applying the external stress signal so that the narrow pass bands of the phaseordered dielectric have maximum sensitivity to the signal. The narrow pass bands Aill at the angles IlI of total transmission are defined in the description of the third embodiment of this invention. The pass bands Ail/ are a function of the effective lattice spacing and the index of refraction of the phase-ordered dielectric. Any externally applied stress, whether it be mechanical or electrical in nature that alters the lattice spacing or index of refraction, may be considered as a suitable signal for the transducer. The claim of this invention is that the transducing medium is a phase-ordered dielectric which utilizes the narrow pass bands, discovered by the applicant, at small grazing angles of incidence. A preferred embodiment of the invention utilizes a collimated, monochromatic source of optical radiation, a slab of the phase-ordered dielectric mounted in free space, and optical detectors of the radiation transmitted and reflected by the dielectric. Another preferred embodiment of this invention utilizes a discrete, phaseordered dielectric which is adjoined by an absorbing detector of the transmitted radiation at the exit surface of the dielectric. For both of these embodiments, the thickness of the phase-ordered dielectric does not appreciably affect the transducer properties of the dielectric. The phase-ordered dielectric may be-of arbitrary geometry with planar or curved surfaces. However, a slab or disc geometry is preferred. BRIEF DESCRIPTION OF THE DRAWING The features of the present invention are better understood from the following descriptions made in conjunction with the accompanying drawing, in which: FIG. 1a FIG. 1e are phase-ordered metallic conductors which, in accordance with the principles of the present invention, serve as conductors of metallic radiation guides of conventional modes of propagation; FIG. 2a is an axial cross-section of a dielectric radiation pipe, made in accordance with the principles of the present invention, for the propagation of electromagnetic waves by successive multiple reflection; FIGS. 2b and 2c are dielectric radiation pipes, of tubular and slab geometries respectively, made in accordance with the principles of the present invention, for the propagation of electromagnetic waves by successive multiple reflection; FIGS. 3a and 3b are electromagnetic band pass filters made in accordance with the principles of the present invention; FIG. 4 is an electromagnetic system, made in accordance with the principles of the-present invention, for the investigation of the lattice structure and lattice defects of materials; FIG. 5 is an electromagnetic mirror made in accordance with the principles of the present invention; FIG. 6 is an electromagnetic transducer made in accordance with the principles of the present invention; FIG. 7 shows phase differences across singlescattered wave fronts; FIG. 8a shows cyclical allowable values of the amplitude delay X in dielectrics; FIG. 8b shows the cyclical path of the vector X as the angle of incidence is varied from 11/2 to O radians; FIG. 9 shows the intensity of the reflection coefficient in the vicinity of the angles of total transmission; FIG. 10a shows multiple-scattered waves in a phase ordered discrete medium; FIG. 10b shows such waves in a continuous medium; FIG. 100 shows such waves in a discrete medium with equivalent parameters of a continuous medium; FIG. 11 shows the intensity, of the reflection coefficient for phase-ordered media subject to the conditions FIG. 12 shows the intensity of the reflection coefficient in the vicinity of the Bragg angles for total X-ray reflection; FIGS. 13 and 14 show the effective relative phase velocity and angle of refraction, respectively, in phaseordered media subject to the conditions set forth above in the description of FIG. 11; FIG. 15 shows electromagnetic properties of a discrete, phase-ordered metal relative to those of a continuous metal; FIG. 16 shows the intensity of the reflection coefficient in phase-ordered media subject to the conditions FIG. 17 shows the H-V phase space in phase-ordered media with zero absorption loss; FIG. 18a shows a semi-unbounded medium j with positive traveling waves; FIG. 18b shows a semi-unbounded medium 1' with positive and negative traveling /waves; and FIG. 180 shows a bounded inedium j with standing waves. DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS Referring now to FIGS. la Ie, the boundary walls of the metallic radiation guides of this invention comprise a phase-ordered metallic conductor 1 which may be of slab, tubular, rod, or other geometry. The conductors 1 of FIGS. 1a 1e may be arranged to form radiation guides of conventional geometry such as coaxial line, two-wire line, wire over a ground plane, unbalanced and balanced strip line, rectangular waveguide, circular waveguide, etc. For purposes of enhancing the mechanical or electrical properties of the radiation guide, the conductors 1 may adjoin dielectric or metallic media 2 and 3 which are not necessarily phaseordered. The metallic radiation guides of this invention are not shown in FIGS. 1a 1e because the geometry of any existing metallic radiation guide with a fixed mode of propagation is applicable to this invention. The claim of this invention is not conductors l but rather the use of conductors 1 in existing metallic radiation guides with a defined mode of propagation. With reference to FIG. 2a 20, a phase-ordered dielectric j, which is mechanically supported by a medium 3 and which bounds a dielectric medium i, serves as the reflecting walls of a dielectric radiation guide. The radiation is incident at a grazing angle ill given by 111 s rl; g 11: where 111 and rp are the angles corresponding to boundaries of total reflection regions at sufficiently small grazing angles of incidence. The radiation propagates in the axial direction in medium i by multiple reflection at the grazing angle II: as shown in FIG. 2a. Dielectric radiation guides of circular and slab geometries respectively are shown in FIGS. 2b and 2c. With reference to FIGS. 3a and 3b, a phase-ordered dielectric j serves as an electromagnetic band pass filter at certain small grazing angles of incidence 111 111 Except for radiation incident at the angles 11: of total transmission, the incident radiation d is totally or almost totally reflected in the reflected direction 2 For ib zli (and for the corresponding wavelengths-A z A the incident radiation a is totally or almost totally transmitted in the forward direction f. In FIG. 3a the radiation d is transmitted in free space to a detector. In FIG. 3b the radiation d is transmitted to an absorbing detector 2 which adjoins the dielectric j. I With reference to FIG. 4, electromagnetic radiation, 7 from a collimated source l,is incident at grazing angles of incidence tfl lll on a dielectric specimen j whose. lattice structure is being investigated. If 111mm M i- 1,: 2, where 41 are the angles of total transmission for perfect phase order of the-specimen j at small grazing angles of incidence, at least some of the incident radiation will be transmitted in the forward direction to a detector 2 even if the specimen does not have perfect phase order. The amount of radiation and the wavelength which are detected by detector 2 is a measure of the lattice defects. and lattice spacing of the dence i1: is given by 111 s III s di where ill and I115, are the angles corresponding to boundaries at total reflection bands at sufficiently small grazing angles of incidence, then the dielectric j serves as a mirror, with the incident radiation being reflected at the angle 4! in the direction e. With reference to FIG. 6, electromagnetic radiation 4', from a collimated monochromatic source 1, is incident at a grazing angle of incidence ti! di on a phaseordered dielectric j where 4%, M=.tl 2, are the angles of total transmission at sufficiently small angles of incidence. In the absence of any other signal, the incident radiation d is transmitted by the dielectric j in the forward direction f to a detector 2 with little if any radiation being detected at the angle til in the reflected direction e by the detector 3. However, if an external mechanical or electrical signal stress is applied, then the amount of radiation detected by detector 3 in the reflected direction will increase at the expense of the amount of radiation transmitted to detector 2 in the forward direction. The changes in radiation levels at detectors 2 and 3 can be correlated to improve the signal to noise ratio. I There has been described novel systems and components for the utilization of electromagnetic waves in discrete phase-ordered media, particularly in phaseordered metals at arbitrary angles of incidence and in phase-ordered dielectrics at small grazing angles of incidence. It is evident that those skilled in the art may now make numerous uses and modifications of and departures from the inventive concepts- Consequently, the invention is to be construed as embracing each and every novel feature and novel combination of features present in or possessed by the apparatus and techniques herein disclosed and limited solely by the spirit and scope of the appended claims. APPENDIX A ABSTRACT The multiple-scattering and interference of waves, scattered by each scattering center of the medium, is determined for the entire electromagnetic spectrum by modification of the single-scattering coefficients in Darwin's difference equations. Anomalies, created by the use of Maxwells field equations for continuous media, are resolved. The phase difference parameters V= 21rd(sin)/A and E 21rd(1-v )/(2)tsin1l1) are introduced to account for the single-scattering phase difference between two successive lattice planes in the reflected and forward directions respectively. (d lattice spacing parallel to the interface of the incident wavefront, tb grazing angle of incidence, incident wavelength, v relative wave phase velocity for continuous media). Whereas Maxwells field equations are strictly applicable only for the coordinate (0,0) in (H, V) phase space, the independent, unified electromagnetic theory of this paper is applicable to any point in (F, V) phase space including the coordinate (0,0). The parameter V was originally introduced by Bragg to explain x-ray reflection but the significance of the parameter E has never been reported. A medium is defined to be phase-ordered if 6V l, |617,.,,,,| I where 6V,.,,,, and 8m are the root-mean-squared deviations from the average value in the medium of the respective parameters. Phase-ordered media therefore include continuous media, perfect crystals with uniform lattice spacing parallel to the interface, amorphous media for which Vaveme '1, E 1, and artificial lattices such as periodic-loaded transmission lines and media. The results are applicable to any wave (electromagnetic, elastic, wave-mechanical De Broglie wave) in phase-ordered media. Semi-unbounded and bounded media are treated. The effective wave phase velocity in discrete media is dispersive, birefringent, and a periodic function of the single-scattering coefficients. Snells law of refraction and Fresnels coefficients of reflection and transmission are a special case of a more general formulation. The critical angle for total reflection and Brewsters angle for total transmission are characteristic of only one of several total reflection and total transmission bands in phase-ordered media. At low grazing angles of incidence, the transmission bandwidth (Alt A) z 10, 10 and 10 for microwave, visible, and x-ray wavelengths respectively. For grazing angles of incidence smaller than the critical angle for total reflection, partial or total transmission can occur. This result may be related to the transmission loss in clad optical fibers. The surface resistivity of metallic waveguides and the reflectivity of metallic radiation pipes are significantly less for phase-ordered metals than the values that have been determined on the basis of Maxwells equations. The theory 'of the propagation of x-rays, for any arbitrary angle of incidence, is presented. The anomalous x-ray absorption loss at low. grazing angles of incidence is resolved. The theory for bounded media follows easily from the concept of self-consistent, positive travelling, forward and reflected waves at every lattice plane in semiunbounded media instead of from impedance concepts. The conditions for invisibility, total reflection, and extinction in phase-ordered media are presented. Separate single-scattering coefficients are given for ultrathin films. The differential wave equation for continuous media and the difference wave equation for phase-ordered media, although not strictly analogous, are identical in the limit of zero lattice spacing. The theory of multiple scattering in phase-ordered media is related to the more general class of problems of radiative transfer including transport phenomena. 1.0 INTRODUCTION With the notable exception of Darwins difference equations for Bragg'reflection at x-ray wavelengths, the present theory of the propagation of electromagnetic waves in media is based on a second order differential wave equation derivable from Maxwells field equations. Whereas Darwins difference equations assume discrete media consisting of equally-spaced lattice planes, Maxwell's field equations are based on the assumption of continuous media. Continuous media are completely characterized by the two tensor parameters, permeability [1.1.] and permittivity [e], which reduce to scalars for isotropic media. However, classi cally speaking, it is well known that media are not continuous but are discrete and consist of individual atoms or sub-atomic particles. Each atom is a scattering center of the incident radiation. The resultant transmitted and reflected radiation is determined by the interference of the waves scattered by each scattering center of the medium. If the incident wavelength is very much larger than inter-atomic distances and if the medium consists of enough atoms (or lattice planes) so that the averaging of the,discrete scattering events is meaningful, then Maxwells equations are approximately valid because the discrete medium may then be approximated by a continuum of scattering centers with zero phase difference between the waves scattered by adjacent scattering centers. As a consequence of approximating discrete media by a continuum, Maxwells field equations do not directly relate to the single scattering event of each electron in the medium. For examp le, Snells law of refraction and Fresnels" coefficients of reflection and transmission at the interface between two unbounded media have never been formally derived from the single scattering of each electron. For the same reason, physical phenomena, such as Brillouin and Raman scattering, which are associated with pertubations or changes in the quantum states of the scattering centers, are also not directly related to Maxwells equations. Furthermore, Maxwells equations are grossly inaccurate whenever the previously stated conditions for approximating a discrete medium by a continuum cannot be satisfied. For example, Maxwells equations are inadequate for x-rays incident at an arbitrary angle of incidence, for electromagnetic radiation incident at near-zero grazing angles of incidence, and for wave propagation in metals. In these cases, the wavefront phase difference between lattice planes or scattering centers is appreciable. Maxwells equations can also be grossly inaccurate in the treatment of ultra-thin films and of inhomogeneties in a medium particularly if the number of scattering centers near interfaces is comparable to the total number of scattering centers in the medium. Therefore, the present state-of-the-art of electromagnetic theory contains several anomalies caused directly by the use of Maxwells field equations which are based on the assumption of continuous media. The purpose of this paper is to remove these anomalies by the use of a more exact,'unified theory based on the assumption of discrete media. The unified aspect of this paper is achieved by determining the interference of the waves scattered by each electron in the medium. In essence, the contribution of this paper is the successful application of Darwins difference equations to the entire electromagnetic spectrum by suitable modification of the single-scattering coefficients of the equations. For the most part, the theory is not restricted to crystals of long-range order with equal spacing between lattice planes but is also applicable to media consisting of amorphous solids, liquids, gases, and plasma. In particular, this paper demonstrates that the wave phase velocity in discrete media is not equal to (5)0" but instead is a periodic function of the singlescattering coefficients. It is shown that the present form of Snell's law of refraction and Fresnels coefficients of reflection and transmission are a special case of a more exact general formulation. It is shown that the critical angle for total reflection and Brewsters angle for zero reflection are characteristic of only one of several total reflection and total transmission bands in semiunbounded, discretemedia. A theory for bounded, discrete media is developed. The results for discrete media are shown to reduce to those for continuous media as the spacingbetween scattering centers (or lattice planes) approaches zero. The mathematics of discrete media differs significantly from that of continuous media. In continuous media, continuous fields are defined at all points of space and are governed by differential field equations. In discrete media, electric and magnetic fields are defined only at points devoid of interacting matter such as in free space or in the space between lattice planes. In discrete media, waves are treated as though they propagate in free space except for scattering by interacting particles. The propagation of waves in discrete media is governed by difference equations. The difference equations and single-scattering coefficients, which are developed in this paper, can be applied to any wave in much the same manner as the wave equation for continuous media. However, unlike the theory for continuous media, the theory for discrete media is directly related to the physical phenomena as sociated with the single-scattering event of each particle in the medium. Furthermore, the difference equations leading to the wave equation for discrete media are self-consistent in the sense that the parameter being determined (e.g. the electric field intensity) interacts with itself rather than with another parameter (e. g. the magnetic field intensity). Because of these unique features, the theory of this paper is also expected to provide some unifying aspects to problems concerning elastic waves, wave mechanical De Broglie waves, radiative transfer, and transport phenomena. 2.0 BOUNDARY CONDITIONS AND EQUATION Qualitative Theory 7 For waves incident on a semi-unbounded continuous medium (see FIG. 2b), the boundary conditions consist of equating the phases of the incident, reflected, and transmitted waves at the interface (contains Snell's laws of reflection and refraction) and of equating the sum of the amplitudes for the tangential components of the incident and reflected waves at the interface to the amplitude of the tangential component of the transmitted wave (results in Fresnels coefficients of reflection and transmission). However, there are two other boundary conditions which are usually not stated but are always assumed for continuous media. It is assumed that the scattering of waves within the medium is totally constructive for any direction, wavelength, and polarization of the incident'radiation. It is also assumed that if one can define or calculate a permeability and permittivity for a medium which extends to all of-space, then the same values (or even the concept) of permeability and permittivity are valid for a bounded medium. In a continuous medium, both of these assump- THE WAVE tions are correct. The phase difference, between scattering centers in a continuous medium, is zero. The environment at a point near an interface is identical to the environment at a point inthe interior of a continuous medium. For a discrete medium, neither of these assumptions are correct. Nevertheless, if the number of lattice planes (or atoms) near an interface are small compared to the total number in the remainder of the medium, then we shall make the approximation that an effective permeability and permittivity can be defined for a semiunbounded or bounded medium. However, in discrete media, the interference of scattered waves generally cannot be neglected. Therefore, ifa smoothed-out effective permeability and permittivity can be defined for a discrete medium,'they must be functions of the direction, wavelength, and polarization of the incident radiation. As the distance between scattering centers approaches zero and the destructive interference of the scattered waves becomes negligible, the effective permeability and permittivity of a discrete medium must reduce to those for a continuous medium. However, the concept of an effective permeability and permittivity in a discrete medium is at best pedagogic and is useful only if one wishes to compare or substitute into existing results for continuous media. The primary parameters which characterize a discrete mediums interaction with waves are the single-scattering coefficients of the medium. Any other parameters which characterize the mediums wave interaction, such as an effective permeability and permittivity, are secondary parameters and are derivable from the single-scattering coefficients. Before determiningthe specific single-scattering coefficients for electron scattering of electromagnetic waves (see Section 3.0), it is useful to obtain some preliminary qualitative results of the interference of waves in discrete media. The following statements, which are supported in Section 3.0 are helpful in understanding the qualitative results. If a plane monochromatic wave is incident on a single electron, then the electron scatters a portion of the incident wavefront in all direction's. lf the same monochromatic wave is incident on an atom containing 2,. bound electrons, then the interference of the waves scattered by each electron is not totally constructive (i.e. is not 2,, times the wave scattered by a single bound electron) but instead is partially destructive because of the non-zero distance between electrons. If the same monochromatic wave is incident on a plane array (lattice plane) of the same atoms, then a portion of the wavefront is transmitted in the forward direction without scattering, a portion is scattered (after undergoing aphase change of 1r/2 radians)v in the same forward direction, and the remainder of the wavefront is scattered (after undergoing a phase change of rr/2 radians) at a grazing angle of reflection equal to the grazing angle of incidence. It is assumed that the lattice plane is of sufficiently large extent so that F raunhofer aperture diffraction can be neglected. It is assumed that the scattering particles may be in thermal agitation but that diffuse Brillouin scattering can be neglected. It is also assumed that the scattering cross-section of the particles is-sufficiently small so that the mutual interactions between scattering centers can be neglected. if several lattice planes composed of similar scattering particles are placed parallel .to the first array at a distance d from it, theninterference occurs between the forward and reflected waves from each lattice plane. it will be shown in the quantitative treatment of this section that the interference of the scattered waves from every lattice plane of the medium determine the refractive properties of the medium. However, a qualitative appreciation for the interference of the waves may be achieved if the interference between waves scattered by the atoms of only two adjacent lattice planes is considered. The interference of waves within the atom will also be neglected for the present time. With reference to FIG. 7, consider that portion AA of the incident plane wavefront of wavelength whose rays are between those which intersect the z axis of the lattice planesat B and B. The distance BB d is the separation between lattice planes. The incident wavefront AA' is of uniform phase normal to the direction lb of incidence. The reflected wavefront CC, consisting of only the two rays BC and BC' normal to the direction ill of reflectance, is not of uniform phase but instead has a phase difference across its wavefront of 2V =(21r/A) (AB'C' ABC) (Zn/ll) (D'B BE) (Zn/A )2DB (Zn/k )2d sin 11:. The phase difference V across the wavefront BB is given by where V is a real number. The wavefront amplitude delay D corresponding to the phase difference V is given by Bdur physical conditions of particular interest, defined by the phase V, are V= 0 (Condition satisfied by a continuous medium). V 1 (Condition for most wave-interactionswith real media over most of the electromagnetic spectrum). V= (2N l)(rr/2), N l, 2, 3, (Condition for total transmission if the refraction of the medium could be neglected and if the interference of waves scattered by the electrons comprising a single atom could be neglected). V= Nrr, N l, 2, 3, (Condition for Bragg reflection at x-ray wavelengths, neglecting the interference of waves scattered by the electrons comprising a single atom and neglecting the refraction of the medium). The phase V is tabulated in Table l for electromagneticv waves in some solid materials. The three grazing angles of incidence which are considered in Table l are T-ABLB ti -THE PHASE v iron ELEo'iEoiiAEr iaiiowAvEs IN soLID MATERIALS [V= (2111A) sin w (radiansfl 1 Ordinary form. 17/2 normal incidence iii, =sin (N \/2d) Bragg angles for x-ray reflection, N= 1,2, 3, ill, sin 1 v critical angle for total reflection in continuous media for v s l where v v,-/v, relative wave phase velocity for continuous media i andj. It will be noted that O 1 over mowof fhe spectrum. Yet, for this considerable portion of the spectrum, no theory exists except the approximate theory based on the assumption V= 0. At x-ray wavelengths for angles of incidence appreciably larger than the critical angle w the phase V satisfies; the condition V 1. Yet, for this portion of the spectrum, adequate theory exists only for angles of incidence in the vicinity of the Bragg angles At x-ray wavelengths, for angles of incidence approximately equal to or less than the critical angle th the phase V satisfies the condition V 1. Yet, no theory exists for these angles of incidence at x-ray wavelengths except the approximate theory based on the assumption V= 0. At x-ray wavelengths far from an absorption edge, sin be is proportional to the wavelength so that the phase V= 21rd(sin tli )l A: V, is independent of wavelength as shown in Table 1. 1 In addition to the interference of rays scattered in the reflected direction all, which is characterized by the phase difference 2V, the interference of rays scattered in the forward direction must also be considered. With reference to FIG. 7, the distance that a ray travels in the forward direction between adjacent lattice planes is given by BF D'F' d/sin ill. However, the transmitted wavefront FF normal to the forward direction is not of uniform phase as is the incident wavefront 8D. The rays in the vicinity of BF and DF of the incident wavefront BD are scattered in the forward direction at B and B respectively whereas the other rays of the wavefront are not scattered. The non-scattered rays undergo a phase change Zrrd/Asiml: whereas the scattered rays undergo an additional phase change (of approximately 17/2 radians as shown in Section 3.0). The average phase difference 217 of the rays in the wavefront FF is not known but may be estimated if it is assumed to a first order approximation that half of the rays are not scattered and the other half are uniformly scattered as though they propagated in a continuous medium. if these scattered rays did not change direction, then their phase change in travelling a distance d/sintll would be 21rd v,,/ A sin 1!: where v w/v, is their wave phase velocity in medium 1' relative to that in medium j if media 1 i and j were continuous. However, the rays of a continuous medium are refracted in a direction lil given by cos 1!], cos liI/Vu. The effective phase change of the scattered rays of the wavefront FF in the forward direction is therefore (21rd v A sin r11)(cos til/COS 41,) 21rd v F/Asintb. The average phase difference 217 of the unscattered and scattered rays in the wavefront F 'F is given approximately by 2L7 2 1rd(l v sintli so iiQ2510)ll/slllikillflfui The phase H is a measure of the refraction caused by a single lattice plane. The phase H is related to the phase V by Tour physical conditions of particularinfere sidefined by the phase u, are: H 0 (Condition satisfied by a continuous medium). [E] 1 (Condition for most wave interactions with real media over the electromagnetic spectrum). 17] (2M 1) 11/2, M O, i l, i 2, i 3, (Condition for 7 total reflection if multiple scattering between lattice planes and the interference of waves scattered by the electrons comprising a single atom could be neglected). E M1r,M 0, :L l, :t 2, i 3, (Condition for total transmission if multiple scattering between lat-. tice planes and the interference of waves scattered by the electrons comprising a single atom could be neglected). in Table 7, (see p. 62a) the phase it is tabulated for free space-metal interfaces for any arbitrary angle of incidence at wavelengths longer than optical wavelengths. For the better conducting metals, IHI l even for normal incidencenPresent theory for the propagation in metals is based on the assumption 7 O. The phase H is also tabulated in Table 2 for electromagnetic waves in free space which are incident on quartz at grazing angles of incidence given by rr/2 normal incidence iii, sin V l v critical angle of reflection I V l i ,M sin" [d (l v )/M)\], M= 0,21,: 2, 1 -3, qualitative sion it will be noted that 0 |fi| l over most of the electromagnetic spectrum for a free space-quartz interface. Yet, for this considerable portion of the spectrum, no theory exists except the approximate theory based on the assumption H 0.At grazing angles of incidence sufficiently less than the critical angle 11:, for total reflection, the phase E satisfies the condition IE I I. No adequate theory exists for thesegrazing angles of incidence. Fresnels equations, based on the assumption u 0, predict that total reflection should occur for 41s til v 4 1, lm v 0. However the condition condition for total transmis- TABLE 2. THE PHASE 1 FOR ELECTROMAIiEgqI/J WAVES lNCiDENT FROM FREE-SPACE ONTO NoTE.(l=4.!) i0' m. vn=lfl5 (microwaves), 1.5 (visible, infrared), 2.0 (ultrnvlolot). H (2M l)1r/2 which corresponds to totally destructive interference states that total reflection shall occur at grazing angles 111 given by (l v )/sin 1,l1 ={(2M+1)1r/V M= 0, $1,: 2, :3, Im v 0, multiple scattering and the interference of waves scattered by the electrons comprising a single atom are neglected. (2.5) Equation 2.5 implies that total reflection can occur at discrete angles for v 1. Equation 2.5 also implies that total reflection does not occur for all angles less than the cntical angle (li for v 1. With reference to Table 2, total transmission occurs at a grazing angle in, 1 10* radians for visible radiation of wavelength A 5 X m incident on a free spacequartz interface. This result is in contrast to the almost total reflection predicted by Fresnels equations. With a well-collimated light source and a sufficiently large sample so that Fraunhofer diffraction effects are negli gible, total transmission should be able to be observed at approximately this angle. However, it will be shown in Section 4.0 that thewidth of the transmission band is extremely small and is smaller than the reflection band for Bragg reflection. Whereas Bragg reflection requires materials whose thickness is at least of the order of the penetration depth and whose lattice structure is of correspondinglong-range order (x-ray penetration is of the order of 1000 lattice planes), total transmission at low grazing angles of incidence is enhanced as the thickness of the material is reduced because the number of scattering centers is reduced. For total transmission, the lattice structure need only be of short-range order equal to the depth of penetration. If the medium were unbounded, then total transmission would occur only if its lattice structure were of long-range order extending to infinity. Narrow transmission bands at low grazing angles of incidence and total reflection bands for v 1 are expected to find application in the investigation of the structure of materials and in the propagation of surface waves, particularly in thin films and fibers for precise filtering and selection of the direction or wavelength of the incident radiation. The Bragg angle for total reflection at x-ray wavelengths and the angles for total reflection and total transmission given by Equations 2.5 and 2.6 are based on assumptions of the nature .of the wavefronts CC and FF in FIG. I, particularly in regard to the number of scattered and non-scattered rays in each'wavefront. in Sections 3.0 and 4.0 an independent, exact, quantitative treatment, free of such assumptions, gives results which are in substantial agreement with the preceding qualitative results. I The phases V and E are both functions of the lattice spacing d parallel to the interface. However, most materials are polycrystalline or amorphous and do not have a well-defined lattice structure of long-range order. Furthermore, many crystals do not have a lattice structure in which the atoms are arranged in equally spaced planes parallel to the interface and therefore cannot be characterized by a single lattice spacing d. Ewald has circumvented this latter difficulty by introducing the concept of reciprocal vectors. The problem of treating polycrystalline and amorphous materials has remained unsolved and has been a severe limitation in many other fields of physics besides electromagnetic theory. The following method for treating polycrystalline and amorphous materials is now offered. The interference of secondary waves which are multiple-scattered is generally dependent upon the order in which they interfere and the number of scattering events. For these reasons the effective phase interference parameters V and E of a polycrystalline or amorphous medium generally are not equal to the average phase parameters V,,,. 21rd, .(sin rid/A and Hi 21rd, (l v,, 2 A simla) where d the average spacing of the lattice planes parallel to the interface in a polycrystalline material or the average spacing between neigh boring atoms in an amorphous material. However, there are conditions for which the resultant interference in polycrystalline and amorphous materials is relatively unaffected by the use of the average phase interference parameters V and H For example, if the spacing d for each pair (m,n) of adjacent lattice planes in a polycrystalline material or of atoms in an amorphous material satisfied the conditons IV V because the interference of the phase differences are constructive. However even if a small percentage of the spacings d do not satisfy the above conditions, the resultant interference is still relatively unaffected assuming that the scattering cross-sections of the atoms associated with the spacings d are not appreciably larger than the majority of the scattering centers. Therefore let the root-mean-squared deviation from the mean of the parameters d, V, and E be given res ectively by rma V un mn) i rma ar mll) 9 rml (H E F. The effective phase difference parameters in polycrystalline and amorphous materials may therefore be estimated to a first order of approximation, subject to the appropriate conditions. by V z al-1 rmf 1 a z am lwml l (2.7) A material which satisfies Equations 2.7 is defined as phase-ordered. If Conditions 2.7 are satisfied by polycrystalline or amorphous materials, the problem of obtaining approximately valid results with such materialsreduces to the problem of treating a single crystal for any value of the phase parameters Vand 17. Before concluding the qualitative theory, it is useful to state some properties regardingthe phase order of materials. Continuous media and perfect single crystals, whose atoms are arranged in equally spaced planes parallel to the interface, satisfy the conditions 8d 8V,,,,,= 8t7,,,,,= 0. Wave interactions, in materials which satisfy the conditions V l, i 1, also satisfy the conditions 8V,,,,, 1, 16a 1 but the converse is not true. For a solid amorphous material, the average spacing d between neighboring atoms is determined by assuming that each atom (or molecule) occupies a volume (4 1r/3)(d, ./2) A/pN where A atomic (or molecular) weight (gms), p material density (gm/cm and N Avogadros number 6.022 X 10. In Table 3 the stfuctured vsfamorphou's spacings of some solid materials are compared. Most liquids can be treated as amorphous materials. However, some liquid solutions should be treated as polycrystalline materials since they have a well-defined lattice structure of Patent Citations
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