RADIATION ACOUSTICS
RADIATION ACOUSTICS Leonid M. Lyamshev
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RADIATION ACOUSTICS
RADIATION ACOUSTICS Leonid M. Lyamshev
CRC PR E S S Boca Raton London New York Washington, D.C.
Library of Congress Cataloging-in-Publication Data Lyamshev, Leonid M. Radiation acoustics / Leonid M. Lyamshev. p. ; cm. Includes bibliographical references and index. ISBN 0-415-30999-9 (alk. paper) 1. Sound-waves. 2. Sound—Transmission. 3. Radiation sources. I. Title. [DNLM: 1. Radiation. 2. Acoustics. 3. Radiation Effects. 4. Thermodynamics. WN 100 L981r 2004] QC243.L93 2004 534—dc22 2003070031
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FOREWORD
Radiation acoustics is a new field of research developing at the interface of acoustics, nuclear physics, high energy physics, and the physics of elementary particles. It is founded upon studies and applications of radiation-acoustic effects arising in the process of interaction of penetrating radiation with matter. The thermoradiation mechanism has been the best studied among the mechanisms of sound excitation by penetrating radiation in condensed media. According to this mechanism, sound generation is caused by thermal expansion of a medium, and the acoustic field can be described as a rule within the framework of linear theory. The book considers mainly the processes of thermoradiation sound excitation in the case of continuous (modulated) and pulsed action of penetrating radiation on a substance. Basic laws of formation of acoustic signals are established and the bonds between the characteristics of these signals, radiation parameters, and thermodynamic, radiation, and acoustic properties of substances are revealed. The efficiency and optimal conditions of thermoradiation sound generation are considered. The particular features of sound generation by a particle beam moving along the surface of a liquid or solid at subsonic and supersonic velocities and an arbitrary form of modulation of radiation intensity in the beam are described. The possibility is discussed of the creation of virtual radiation-acoustic sources of sound in a wide frequency range (from sound to hypersound frequencies) with controlled parameters in liquids or solids. We consider the particular features of thermoradiation generation of sound by single particles. Experimental results on sound excitation by beams of photons (laser radiation), electrons, protons, heavy ions, X-rays, and gamma-quanta are given. Some other mechanisms of sound generation
by single particles in the process of their absorption in a medium are considered apart from the thermoradiation mechanism, i.e., the mechanisms of microshock waves and the bubble, dynamic, Cherenkov, and striction mechanisms. Applications of radiation acoustics are discussed. We have not tried to go into the details of many of them. Our purpose is to demonstrate the prospects of application of radiation acoustics to various fields from microelectronics (radiation-acoustic microscopy) to geophysics (neutrinoacoustic sounding of the Earth), and astrophysics (detection of cosmic neutrino and muons of super-high energy by huge acoustic detectors in the ocean). We have not tried to review all papers on radiation acoustics. On the contrary, we have quite deliberately not included in the book the results of studies of nonlinear radiation-acoustic phenomena arising in the process of interaction of powerful radiation beams with matter. Although the role of nonlinear effects in future radiation-acoustic technologies will undoubtedly be essential (targeted action on physical, mechanical and chemical structure of substances, radiation-acoustic destruction of materials, etc.), investigation of these effects still continues. The book may be useful not only to acousticians but also researchers and technicians specializing in adjacent and other fields as well as postgraduates and university students. I am very grateful to G. A. Askar’yan, F. V. Bunkin, and V. I. Il’ichev for many useful remarks on the manuscript. L. M. Bolotova, M. G. Lisovskaya, and B. I. Chelnokov helped me greatly in the preparation of the manuscript for publication. I am deeply grateful to them. L. M. Lyamshev
CONTENTS
INTRODUCTION
1
Chapter 1. PENETRATING RADIATION: GENERAL INFORMATION
7
1. Elementary particles: Fundamental laws of the microscopic world 7 2. Absorption of penetrating radiation in a condensed medium 13
Chapter 2. BASIC MECHANISMS OF SOUND GENERATION BY PENETRATING RADIATION IN CONDENSED MEDIA 1. Mechanisms connected with heat release 2. Thermoradiation generation of sound 3. Initiation of microshock waves 4. Bubble mechanism 5. The Cherenkov mechanism 6. Striction mechanism of sound generation 7. Sound generation in the process of pulsed radiolysis 8. Dynamic mechanism 9. Other mechanisms of sound generation
Chapter 3. THERMORADIATION EXCITATION OF SOUND IN A HOMOGENEOUS LIQUID 1. Equation of thermoradiation generation of sound 2. Reciprocity theorem in acoustics – Solution technique for boundary problems 3. Excitation of monochromatic sound in a liquid half-space with a free surface – The case of undisturbed surface 4. A liquid half-space with large-scale roughness of boundary
23 23 24 28 30 31 33 35 35 38
39 39 42
47 52
5. The case of small unevenness 6. Efficiency of thermoradiation excitation of sound in a liquid – Some estimates
Chapter 4. THERMORADIATION EXCITATION OF SOUND IN AN INHOMOGENEOUS MEDIUM 1. Sound excitation in a liquid half-space in the presence of a layer of another liquid at its boundary 2. Generation of sound in a liquid adjoining a solid layer 3. Liquid half-space with an inhomogeneous surface layer
Chapter 5. EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES 1. Sound generation in a liquid by rectangular pulses of radiation 2. A liquid with rough surface 3. Radiation pulses of arbitrary shape 4. Near wave field of thermoradiation pulsed source of sound 5. Sound generation in a liquid with gas bubbles
Chapter 6. GENERATION OF SOUND IN SOLIDS BY INTENSITY-MODULATED PENETRATING RADIATION 1. Basic equations 2. Boundary conditions 3. Method for solution of boundary problems 4. Thermoradiation generation of sound in a solid half-space with a free boundary 5. Particular features of excitation of Rayleigh waves 6. Solid half-space with a liquid layer at its surface 7. Efficiency of thermoradiation generation of sound 8. Influence of particular features of absorption of penetrating radiation on sound generation
61 72
75
75 88 96
105
106 113 116 124 130
137 137 139 141 147 151 156 160 164
Chapter 7. PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS 1. Sound generation by radiation pulses in a solid half-space 2. Excitation of Rayleigh waves by radiation pulses 3. Sound generation in a solid half-space in the presence of a liquid layer at its surface 4. Efficiency of sound generation 5. Influence of particular features of absorption of penetrating radiation 6. Thermoradiation generation of sound by pulses of non-relativistic protons
Chapter 8. MOVING THERMORADIATION SOURCES OF SOUND 1. Sound generation by a moving thermoradiation pulsed source in a liquid 2. Sound excitation by a moving thermoradiation pulsed source in solids
Chapter 9. SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES 1. Sound generation by a particle in infinite space 2. Sound excitation by single particles in a solid half-space 3. Particular features of excitation of Rayleigh waves 4. Efficiency of sound generation
Chapter 10. EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION OF SOUND 1. Laser thermooptical (thermoradiation) sources of harmonic sound oscillations in water 2. Sound excitation in water by laser pulses 3. Sound field excited by a sequence of laser pulses 4. Acoustic field of a moving laser thermoradiation source of sound in water
169
169 184 186 191 193 195
201
202 213
227 227 234 237 238
239
240 248 252 257
5. Laser thermooptical excitation of sound in solids – Excitation of surface waves 6. Sound excitation by X-rays (synchrotron radiation) in metals 7. Sound excitation by a proton beam 8. Excitation of acoustic waves in metals by electrons, positrons, and γ-quanta 9. Sound generation by an electron beam in water 10. Sound excitation by a beam of ions in metals
Chapter 11. SOME APPLICATIONS OF RADIATION ACOUSTICS 1. Scanning radiation-acoustic microscopy and visualization 2. Scanning laser-acoustic microscopy 3. Scanning electron-acoustic microscopy 4. X-ray – acoustic scanning visualization 5. Ion-acoustic microscopy and visualization 6. Acoustic detection of super-high-energy particles in cosmic rays – The DUMAND project 7. Neutrino for geoacoustics – The GENIUS project
270 281 286 298 302 306
309
310 311 314 318 321 325 330
Conclusion
339
References
343
Addendum. ACOUSTOOPTICS OF PENETRATING RADIATION
359
1. Diffraction of X-rays and neutrons by ultrasound in crystals 2. Scanning acoustic tunneling microscopy 3. Interferometers using matter waves – Atom interferometers
SUBJECT INDEX
360 361 362
367
INTRODUCTION
Investigation of radiation-acoustic effects was stimulated mainly by progress in the field of high-energy physics and the physics of elementary particles. The latter has advanced greatly during recent decades. Particles of tremendous energy of the order of magnitude of tens, hundreds, and thousands of gigaelectronvolts (GeV) were obtained with the help of accelerators and many new elementary particles subjected to amazing interactions and inter-transformations were discovered. Quantum chromodynamics and unified theory of electromagnetic and weak interactions appeared. The state-of-the-art is now that physics is on the verge of creating the unified theory of all the fundamental interactions — electromagnetic, strong (nuclear), weak, and gravitational interactions. Experimentation at even greater energy is needed for solution of this problem. This needs powerful accelerators of elementary particles, which would provide an opportunity to make the next step into the depth of the microscopic world [2, 76, 89, 176]. Such accelerators are under design and construction now. As the construction of new, more powerful accelerators opens new opportunities for advancement of investigation of elementary particles into the field of larger and larger energies, the accelerators designed initially for purely basic studies, are applied to a greater and greater extent in research into the physics of solids, biology, chemistry, and medical science. They are utilized successfully in radiation technology, defectoscopy, analysis of rare minerals, and also (as it will be demonstrated below) in radiation-acoustic research and technology. As a rule, these are low-energy (about several megaelectronvolts (MeV)) accelerators like betatrons, linear accelerators, and microtrons [13]. Now more powerful accelerators (up to several 1
2
INTRODUCTION
gigaelectronvolts) of proton, meson, and ion beams and X-rays are being tried for these purposes. The beginning of radiation acoustics is connected in a broad sense with the discovery by A. Bell, W. Roentgen, and J. Tyndall [180, 195, 245] of the optoacoustic (photoacoustic) effect, that is sound generation in a gas volume due to intermittent (modulated) light passage or, in other words, due to interaction of modulated optical radiation (modulated photon beam) with a substance (gas). At the same time Bell discussed the problem of construction of a radiophone, “a device for producing sound by radiation of any kind” [195]. Further studies of the optoacoustic effect and its applications served, as is known, as the basis for the development of optoacoustics (photoacoustics) including optoacoustic spectroscopy of gases and condensed media [74, 126]. A powerful stimulus for the development of this field in recent decades was the construction of lasers (see [96, 127] for example). The first studies of radiation-acoustic effects were conducted in the 1950s and 1960s. So, for example, in 1956 Kaganov, Lifshits, and Tanatarov considered sound radiation in a solid by a uniformly moving electron and showed that at an electron velocity greater than the sound velocity in a medium (lattice), the Cherenkov radiation of sound (phonons) occurs [101]. The analogous problem was considered earlier (1953) by Buckingham [203]. In 1955 Glaser and Rahm reported observation of tracks of particles in the process of their passage through a metastable boiling-up liquid in a bubble chamber according to sound (vibration) signals arising as a result of the birth and development of bubbles [209]. In 1957—1959 Askar’yan considered radiation of ultrasonic and hypersonic waves by charged particles in dense media due to local heating and formation of microscopic cavities along particle tracks. Excitation of surface and bulk waves under the impact of a non-relativistic electron flux upon the surface of a dense medium was considered and the problem of utilization of acoustic signals generated by particles for detection of particles was discussed [6, 7]1. In 1963 White investigated sound generation by an electron beam in a solid [260]2. Somewhat later (1967) Graham and Hutchison [216] measured 1
In contrast to Glaser and Rahm [209], who discussed sound radiation due to the rise of bubbles at particle tracks in metastable media, Askar’yan [6, 7] considered local heating arising at particle tracks in dense stable media and producing sound pulses, as well as the bubbles generating hypersonic waves. He proposed also acoustic detection of particles and noted the possibility of manifestation of hypersonic pulses in the process of biological action of radiation on cells and chromosomes (as a part of destructive effect). 2 We must note that at the same time, White conducted the first experiments on laser generation of sound in solids. Somewhat earlier the first widely known experiments
RADIATION ACOUSTICS
3
mechanical oscillations in quartz crystals, sapphire, etc., on their being irradiated by electron beam pulses, and in 1969 Beron and Hofstadter, as Askar’yan before them [6, 196], suggested that not only electrons but also other particles can generate mechanical vibrations3. Numerous studies of sound excitation in condensed media by electron and proton beams and by single particles were conducted in the 1970s by Borshkovskii, Volovik, Zalyubovskii, Kalinichenko, Lazurik, and others, and in the 1980s, by Lyamshev and Chelnokov (see [155, 156]). Various mechanisms of sound excitation in condensed media by penetrating radiation were considered. The major results of these studies and the bibliography can be found in the book [97]. The publications of many researchers on possible applications of the radiation-acoustic effects date to the same period (see, for example, F. Perry et al. on the application of these effects to the dosimetry of pulsed beams of accelerated particles and to obtaining data on the depth distribution of irradiation dose in a target [242]). A powerful stimulus for development of research in radiation acoustics were ideas to use radiation-acoustic effects for detection of super-highenergy muons and neutrinos at a large depth in the ocean [8, 199], to develop a radiation-acoustic microscope [207], and finally, the suggestion to “sound” the Earth (using a radiation-acoustic technique) by a super-highenergy neutrino beam from super-powerful (for super-high energy of particles) proton accelerators of future generation named tevatrons [248]. Further publications (see, for example, [15, 151 – 154, 184]) were connected in this or that way with these aspects. Investigations performed in the 1950s and 1980s have been described to some extent in the book [97] mentioned above and in a collection of articles [173].
on interaction of laser radiation with a liquid were conducted in the Lebedev Physical Institute of the USSR Academy of Sciences (see G. A. Askar’yan, A. M. Prokhorov, G. F. Chanturiya, and G. P. Shipulo, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1963, v. 44, No. 6, pp. 2180 – 2182) which led to the discovery of the light-hydraulic effect. This paper played a fundamental role in the development of laser and radiation acoustics. 3 Recently detection of an acoustic signal from a muon beam at the neutrino channel of the U-70 accelerator of the Institute of High-Energy Physics (Moscow) was reported (see A. B. Borisov et al., Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1991, v. 100, p. 112). Acoustic signals from a muon flux in the muon filter of the neutrino channel were studied. Results of measurements of signals and theoretical estimates based on the thermoradiation mechanism of sound generation were given. A satisfactory agreement between experimental and theoretical results was noted that was the evidence of the dominant role of the thermoradiation mechanism in the process of sound generation. The possibility of application of the radiation-acoustic technique to remote determination of the characteristics of particle beams from the accelerator was considered.
4
INTRODUCTION
A definite concept has been formed by now about the mechanisms of sound generation by penetrating radiation. They are connected usually with the physical phenomena (processes) resulting in the conversion of penetrating radiation energy into acoustic energy. These processes depend on the radiation type, on the target substance in which this radiation is absorbed, and on the energy release mode in the absorption region. The mechanisms of sound generation are numerous and not equal in their efficiency [155]. Heat release is one of the most universal physical phenomena taking place due to absorption of penetrating radiation. Thermal energy can transform partially into sound wave energy in different ways. At moderate released energy densities, when no phase changes occur in a substance, the main contribution to the sound generation process arises from the thermal expansion of a medium. This is the thermoradiation (thermoelastic) mechanism of sound generation. In this case the sound fields can be described within the framework of linear theory which has been developed extensively in recent years [156]. The pattern of sound generation looks much more complex in the case of large densities of penetrating radiation energy released in a medium. In this case the phenomena arising are nonlinear. The effects caused by the growth of the expansion rate of the heated region of a medium (hydrodynamic nonlinearity) and the change of thermodynamic parameters of a substance in the process of action of penetrating radiation (thermal nonlinearity) turn out to be substantial. If the density of released thermal energy increases further, more complex processes of sound generation develop, which are connected with phase transitions, for example, in the conditions of bubble mechanism of sound generation by penetrating radiation [6, 64] and the mechanism of formation of shock waves [4, 72]4. There are also “non-thermal” mechanisms of sound generation: the Cherenkov, dynamic, striction, and other mechanisms. However, the thermoradiation mechanism is the one best-studied up to now. Research on radiation-acoustic effects could hardly have drawn the attention of physicists in recent decades if it was not connected with the prospects of its practical applications. Examples are scanning radiationacoustic microscopy of condensed media [157, 201, 206, 257], acoustic detection of super-high-energy particles (the DUMAND project — Deep Underwater Muon and Neutrino Detection) [9, 56, 67, 200, 239], research 4
We should note that discussion on the density of the energy released in a medium in the process of absorption of penetrating radiation concerns first of all laser (optical) radiation (photon beams). However, in a certain sense they are valid also for particle beams and a single particle or a group of particles. In the latter cases we can discuss the peaks of local heating and overheating leading to formation of acoustic compression waves, microshock waves, microcavitation, and microbubbles.
RADIATION ACOUSTICS
5
on the role of the radiation-acoustic mechanism in underwater noise generation in calm ocean [149], and also the opportunities opening up for the application of new generations of proton and linear super-powerful accelerators of future generations to the production of super-high-energy neutrino beams and the application of these beams in geoacoustics (neutrino geoacoustics, the GENIUS project — Geological Exploration of NeutrinoInduced Underground Sound) [185, 248] and in neutrino-acoustic ocean tomography [236]. We should note also the important role of radiationacoustic effects in investigation and prediction of radiation blistering [80]. The book consists of eleven chapters. The first two chapters give data on elementary particles, absorption of penetrating radiation in a substance, and mechanisms of radiation excitation of sound. The next seven chapters of the book contain the results of the theoretical treatment of thermoradiation sound generation in condensed media, i.e., in homogeneous and inhomogeneous liquids and solids, under the action of modulated penetrating radiation and radiation pulses on a substance. Particular features of the acoustic fields of moving thermoradiation sound sources are considered. Sound excitation by single high-energy particles is analyzed. The efficiency and optimal conditions of thermoradiation sound generation are discussed. The theoretical consideration is based on the solutions of boundary-value problems for the inhomogeneous wave equation with the right-hand side in the form of the function of power density of sound sources produced by radiation absorption in a substance. It is assumed that this absorption obeys the exponential law, which is valid for laser (optical), X-ray, and electromagnetic (in general) beams and with certain limitations for beams of relativistic electrons. This has provided an opportunity to obtain results in the final form and compare them to experimental data. At the same time, the role of the law of radiation absorption in formation of acoustic field of a thermoradiation sound source is analyzed, and the conditions when the absorption law does not play a considerable role are determined. Corresponding analytical expressions are given. Chapter 10 presents the results of numerous experiments conducted and published by many researchers and concerning thermoradiation excitation of sound by modulated laser radiation and laser pulses in the cases of stationary and moving laser beams, beams of protons and electrons in water, and by electron, positron, proton, ion, and X-ray beams and gamma-quanta in metals. We have to note here that comparison of these results to theoretical conclusions proves the validity of the thermoradiation theory. Some applications of radiation acoustics are discussed in Chapter 11. The purpose of this chapter is to demonstrate not only the variety of already existing applications but their “large scale”. We mean both radiationacoustic microscopy and immense projects of the future like the DUMAND and GENIUS projects.
CHAPTER 1
Penetrating Radiation: General Information This chapter provides information on particles constituting penetrating radiation, and absorption of penetrating radiation in the process of its interaction with a substance. Only the most general concepts are presented here. Detailed information may be found in specialized books on particle and nuclear physics.
1. ELEMENTARY PARTICLES: FUNDAMENTAL LAWS OF THE MICROSCOPIC WORLD The material world is “constructed” from elementary particles. This means that their properties, laws of motion, and forces between them determine the diversity of physical phenomena. Commonly the particles which cannot be separated into components are called elementary particles. This definition applies to electrons, protons, and neutrons, but not atoms and atomic nuclei. Protons and neutrons together are called nucleons. Another common and well-known particle is a light particle, i.e., photon. An electrically neutral particle, i.e., neutrino, is much less known. It is very difficult to detect, as it interacts with electrons and nucleons very weakly and therefore, goes through a tremendous thickness of substance almost freely. Knowledge on the structure of the microscopic world, i.e., physics of elementary particles, is the basis for the whole of modern science. Studies of atomic structure
7
8
PENETRATING RADIATION: GENERAL INFORMATION
provided an opportunity to discover extraordinary properties of elementary particles and develop a theory of motion, i.e., quantum mechanics. Quantum mechanics and the theory of relativity are the two pillars supporting the whole of modern physics. Such a general concept as symmetry, which to a significant degree determines the structure of particles and their interactions, is also fundamental for modern physics. Modern models and theories of physics of elementary particles are formulated in the mathematical language of the apparatus of symmetry, i.e., theory of groups. One of the most important parameters in quantum mechanics is spin. All particles are separated into classes depending on the value of their spin: particles with half-integer spins are fermions or Fermi particles and particles with integer spins are bosons or Bose particles. The description of spin using the mathematical apparatus of the theory of groups became the starting point of many theories, i.e., so-called internal symmetries. Development of symmetry schemes unifying fermions and bosons is the goal of the supersymmetry trend and finally, the Grand Unification Theory. Let us turn to history. It was discovered in the first decade of the last century that an atom consists of a nucleus and electrons. As it turned out, a nucleus has the dimensions of 10−13 cm and the whole atomic mass is contained in it. The density of matter is extremely high in a nucleus and is equal to 1014 g/cm3. The charge of the nucleus is positive. Electrons move around a nucleus at a distance of 10−8 cm. It was determined in the 1930s that a nucleus consists of protons and neutrons. The last do not have any charge. Electrons are held within an atom by electric forces. Physicists call the forces binding positively charged protons with neutral neutrons nuclear forces, due to their nature. Pauli had predicted the existence of the neutrino already in the 1920s. It only became possible to observe this particle experimentally twenty years after it had been discovered because of its “ability” almost not to interact with matter. Physicists associated this feature with forces of weak coupling in contrast to nuclear forces, i.e., forces of strong coupling. Further research, and first of all the studies of the nature of nuclear forces, led to the discovery of a huge number of particles, their interaction, and interconversion. Quantum electrodynamics, quantum chromodynamics, and the unified theory of electric weak interaction were developed. All particles are divided into hadrons and leptons depending on processes they take part in. Fundamental interactions of only four types stand behind all processes observed up to now, i.e., electromagnetic, weak, strong, and gravitational couplings. Gravitation is universal. All elementary particles take part in it. Sources of electromagnetic field are charges. Neutral particles, which do not have any charge, interact with an electromagnetic field only due to their complex structure or quantum
RADIATION ACOUSTICS
9
effects. In this sense electromagnetic coupling is not as universal as the gravitational one. The same is also true about weak coupling. As for strong coupling, only hadrons constituting the vast majority of particles (about 200) take part in it. Multiple mesons and hyperons (both long-lived and resonance ones, i.e., with lifetime shorter than 10−20 s) belong to the family of hadrons as well as nuclons. Leptons take part in electric weak coupling and do not participate in strong coupling. There are six of them: electrons e, muons µ, tau-leptons τ, and the corresponding three neutrinos νe, νµ, and ντ1. In contrast to leptons, hadrons may be called elementary particles only in the sense that they are really indivisible. However, it was determined that they have an internal structure and behave as “loose” systems in hadron interactions. Hadrons consist of quarks. According to modern views, quarks are structureless true elementary particles like leptons. As distinguished from hadrons, leptons and quarks are called fundamental particles. However, quarks do not exist separately. They exist within hadrons in a bound state. This property of quarks is called “confinement”. Another, so to say opposite, side of this feature of quarks is the fact that they do not interact when close to each other. This property got the name of “asymptotic freedom”. A hypothesis of existence of quarks was suggested in 1964. It followed from the assumption of the existence of symmetry among leptons and quarks that there should be six quarks. This was confirmed experimentally. In the process of experimenting with high-energy particles, it became possible to “observe” quarks and determine their masses and charges. However, it was impossible (and apparently will not be possible in the future) to knock out a quark from a hadron. Five quarks have been discovered up to now and the search for the sixth one is in progress2. Quark types or “flavors” (as they are called commonly) are denoted by letters u, d, s, c, b, and t. According to theory, each quark must have a certain “color” charge (color is “chromos”; the term “chromodynamics” follows from here). There are three color charges in all. “Color” charges are introduced analogously to common yellow, red, and blue colours. Thus according to theory, there must be 18 quarks and the same amount of anti-quarks. 1
There was very little experimental data on the neutrino until recently. The situation has changed now. Multiple reports on conversion of one type of neutrino into another and non-zero mass of neutrino were published on the basis of studies of solar neutrino and experiments with accelerators. For example, see J. N. Bahcall et al., Nature, 1995, v. 375, p. 29 and A. B. Balantekin, Phys. News, 1995, AIP, p. 49. 2 The discovery of the sixth quark has been reported! See C. Quigg, Discovery of the Top Quark, Phys. News, 1995, AIP, p. 56; F. Abe et al., (CDF Collaboration), Phys. Rev. Lett., 1995, v. 74, p. 2626; and S. Abachi et al., (DO Collaboration), Phys. Rev. Lett., 1995, v. 74, p. 2632.
10
PENETRATING RADIATION: GENERAL INFORMATION
Hadrons are built from them. Hadrons are divided into two big groups: barions with barionic charge and mesons without this charge. Protons, neutrons, and other particles belong to the first group, while the second group includes π-mesons, k-mesons, and so on. Barions are constructed from three quarks with different mutually complementary “color” charges. A meson consists of a quark and an anti-quark. Both barions and mesons are “colorless”. We should stress that in the case of quarks, “color” is just a convenient term to denote quantum numbers characterizing quarks. Color charges of anti-quarks differ from that of quarks. There is a strong symmetry among quarks of different flavors and leptons. Leptons and quarks include three generations of fundamental fermions. Fermions of the first generation together with photons are the construction material for modern matter. As for fermions of the second and third generations, apparently physicists begin to understand now that on the one hand they played an important role at the earliest stages of formation of the Universe and on the other hand our existence depends on the relationship between uand d-quarks and electrons. Studying the nature of the forces binding quarks in hadrons has demonstrated that here lies a deep analogy with electromagnetic forces. Interaction of an electron with another electron or charged particle is performed by photons. They have zero rest mass but do not have an electric charge. Therefore, an electron does not change its charge when it emits a photon. Photons are vector particles. They are described by a vector field (corpuscular-wave dualism). It was demonstrated totally analogously to quantum electrodynamics and its principles of symmetry that interaction between quarks is performed also by vector particles, i.e., gluons, which have zero mass and unit spin like photons. However, unlike photons, there are eight gluons and they have “color” charges corresponding to the laws of quantum chromodynamics. Moreover, they interact strongly with each other forming elementary particles called glubols. Nuclear forces between protons and neutrons (hadrons) in a nucleus are the secondary manifestation of quark-gluon interactions. A theory of quarkgluon interactions has been confirmed experimentally. It has been developed within the framework of quantum chromodynamics and describes formation of “gluon strings” and “quark, anti-quark, and gluon streams” developing when gluon strings get broken in the process of collision of high-energy hadrons. Thus, quarks and gluons (they are often called partons) are fundamental particles constituting hadrons. It was established experimentally that their dimensions are smaller than 10−16 cm, while the characteristic size of hadrons is 10−13 cm. Therefore, partons may be considered point particles like leptons.
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11
The nature of weak forces or weak coupling was described within the framework of the so-called theory of electric weak coupling developed in the 1960s. It was established within this theory that despite the fact that electromagnetic and weak forces are different, manifestations of one and the same coupling, i.e., electric weak coupling, constitutes their nature. According to the electric weak theory, weak couplings are caused by the interchange of intermediate vectors by W±- and Z0-bosons, just as electromagnetic coupling is caused by interchange of photons. In this case the weakness and small radius of weak coupling are explained by the fact that W±- and Z0- bosons are very heavy particles. W±-bosons are charged while Z0-bosons are neutral vector particles. Bosons were discovered in 1983 by European physicists in experiments with an accelerator with counter-propagating proton-anti-proton beams at CERN (Switzerland) and this was a solid confirmation of electric weak theory. It is possible that there exist more of such particles, i.e., intermediate vector bosons. This problem is still open. One of the fundamental properties of weak forces is the fact that they are connected with slow (weak) decays and always associated with the neutrino. Weak forces play an important role in the evolution of the Universe. Nuclear reactions in the Sun are connected with them. The fact that intermediate vector particles (photons, bosons, and gluons) take part in all interactions, successful development of a unified electric weak theory, symmetry among leptons and quarks, etc., indicate the possibility of the existence of a general bond between fundamental forces. The possibility to develop a single theory unifying these forces, i.e., electromagnetic, strong (nuclear), weak, and gravitational forces, follows from here also. The key element of different approaches to formation of the Grand Unification Theory is the development of different kinds of supersymmetry schemes. However, despite the fact that papers devoted to this problem number several thousands already, “traces” of supersymmetry have not been discovered in spectra of known elementary particles. Nevertheless, it is possible that such traces will be discovered in experiments with superpowerful accelerators of new generations. The logic of development of physics of elementary particles is for this3. Let us point out in conclusion the characteristic scales of fundamental forces. Gravitational and electromagnetic forces are long-range. Nuclear 3
It was reported recently that results of very precise measurements obtained lately support predictions on minimal supersymmetric SU (5) model, which unifies electromagnetic, weak, and strong couplings (see U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. B, 1991, v. 260, p. 447; S. Dimoklos, S. A. Raby, and F. Wilczek, Phys. Today, 1991, v. 44, p. 25). It is expected that such “unification” (if it exists) should be observed at the energy E > 1013 TeV.
12
PENETRATING RADIATION: GENERAL INFORMATION
forces act at distances of the order of magnitude of nuclear size. A short range of coupling characterizes weak forces. It is at least two orders of magnitude less than the range of strong coupling. New theories including the Grand Unification Theory need new experiments with more powerful accelerators of new generations4. However, there are fundamental limits in construction of very powerful (“traditional”) accelerators. Estimates show that a hypothetical annular accelerator for energy 107 TeV, which could be constructed using a superconductor magnet, would be a ring with radius about that of the Earth. Even if such an accelerator was located in space, attempts to further increase of its power and size would be impossible because of synchrotron radiation. In the meantime, “natural” accelerators can “accelerate” particles, e.g., neutrino, up to the energy 108 TeV. Methods of radiation acoustics may be useful (or even extremely necessary) for detection of just such particles of super-high energy. Indeed, as will be described in detail in a special section, 7 8 the appearance of particles with energy E ≈ 10 – 10 TeV in the spectrum of cosmic rays is an extremely rare occasion. Probability and frequency of their detection increase with the increase of detector size. Huge detectors (with the size of tens of square kilometers) are needed for this. Just under such conditions, application of acoustic detection methods (i.e., utilization of arrays of hydrophones or seisms) instead of traditional techniques (scintillation counters and Cherenkov detectors in water) can be favorable and justified. Thus, according to modern concepts, six leptons, six types (“flavors”) of quarks, and five bosons (photons, gluons, and intermediate vector bosons γ, g, W±, and Z0) are the fundamental particles forming the world around us. All processes observed up to now are caused by fundamental couplings of four types: gravitational, electromagnetic, strong (nuclear), and weak. We have tried to give the most general idea of the modern state of affairs in the physics of elementary particles5. We do not need a detailed presentation here. Those who are interested in more details can find them in
4
It is useful to note that some theories of supersymmetry connect hopes for 16 development of the Grand Unification Theory with the energy 10 TeV and scales −33 at distances about 10 cm. As it follows from modern concepts, a strict symmetry exists at small distances. This symmetry is violated spontaneously in the process of transition to large scales which leads to the variety of particles and types of their interaction which we observe around us in the process of experimenting with modern large accelerators and in the spectrum of cosmic rays [73, 78,169] (see also the previous footnote). 5 According to a colorful expression by Ya. B. Zel’dovich, a presentation for “pedestrians” [98].
RADIATION ACOUSTICS
13
books by Akhiezer and Rekalo [14], Grishin [78], and Okun’ [168, 169], a paper by Zel’dovich [98], and other papers cited there6.
2. ABSORPTION OF PENETRATING RADIATION IN A CONDENSED MEDIUM In many cases the parameters of the acoustic field generated by penetrating radiation in a condensed medium are determined essentially by the characteristics of radiation absorption in a substance. Absorption depends on the type of particles (quanta), their energy, and the material of a target. Let us consider, in the most general form, some laws of radiation absorption. While moving through a substance, particles interact with its atoms, i.e., electron shells and atomic nuclei (or nuclear nucleons). As we have indicated already, there are three types of interactions involving particles (we ignore very weak gravitation coupling as it is very small and insignificant at this scale), i.e., strong (nuclear), electromagnetic, and weak. In contrast to electromagnetic coupling and in the case of not very large energy of particles, strong and weak couplings are essential only in the case of very small dimensions of the interaction region, about the dimensions of elementary particles and atomic nuclei. Therefore, considering passage of charged particles through a substance, the major role belongs to electromagnetic coupling.
TRANSITION OF HEAVY CHARGED PARTICLES THROUGH A SUBSTANCE Protons, nuclei of various atoms, e.g., nuclei of atoms of helium 4He (α-particles) consisting of two protons and two neutrons, nuclei of hydrogen isotope, isotope of helium 3He (two protons and one neutron), and some other particles taking part in strong coupling belong to heavy charged particles. Heavy charged particles of moderate energy lose it in the process of passage through a substance. This happens mostly because of inelastic collisions with bound electrons of atoms of a retarding substance. When the velocity of a particle becomes so small that it captures electrons, energy losses decrease. However, deceleration of a particle continues until its energy gets reduced down to the thermal energy of substance atoms. The 6
See also papers on accelerators [47, 176, 223, 229, 237, 256].
14
PENETRATING RADIATION: GENERAL INFORMATION
first stage of the process of deceleration, i.e., deceleration during the time when a particle is still charged, may be described theoretically in a satisfactory way [188, 190]. The second stage of the deceleration process, i.e., the stage when a particle captures electrons (e.g., at energy less than 1 MeV for an alpha-particle or 0.1 MeV for a proton), practically cannot be treated theoretically. However, there are reliable experimental data [244]. The loss of energy by multi-charge nuclei has a special feature: they can capture not a single electron but several, which makes the spectrum of energy losses more complex [232]. Theoretical calculation for the first stage of the process of deceleration of a heavy particle leads to the formula [188] dE 4πe 4 z 2 NB , = dx mV 2
(1.1)
where, for example, in the case of relativistic velocity of a particle 2mV 2 − ln(1 − β 2 ) − β 2 ; B = Z ln I E is the particle energy; ze and V are its charge and velocity; M and m are the masses of the incident particle and electron, respectively; N is the number of atoms in the unit volume of a substance; Z is the charge of nuclei of atoms of a medium substance; I is the average excitation energy of an −19 J); β = V/u; and u is the velocity of light. The atom (I = 18.5Z⋅1.6⋅10 quantity dE/dx is called the “stopping ability” of a substance and B is called the deceleration coefficient. Sometimes the quantity B is identified as the “stopping ability” of a substance. In the general case, B is determined either theoretically or experimentally. Passing through a substance, a heavy charged particle performs ionization, which results in loss of energy. The most probable initial collisions with atomic electrons are collisions, such that a relatively slow secondary electron with kinetic energy not exceeding the ionization energy is knocked out. However, secondary electrons with relatively large energy form in the result of a small number of collisions, their maximum energy 4(v/M)E corresponding to the maximum velocity equal to the double velocity of an incident heavy particle. For example, in the case of a proton with energy E = 10 MeV, secondary electrons of different energy may be produced, the maximum energy being equal to 20 keV. These so-called delta-electrons (δ-electrons) ionize atoms of a retarding medium further. Therefore, the initial ionization produced by the charged
RADIATION ACOUSTICS
15
particle itself should be distinguished from secondary ionization produced by delta-electrons. According to experimental data, total ionization exceeds initial ionization approximately three times. Distribution of ionization losses along the path of a particle coincides to a first approximation with the stopping ability of a substance or, in other words, with the distribution of energy losses at the unit length of the path dE/dx. The main result obtained from formula (1.1) is the fact that specific energy losses of a charged particle for ionization are proportional to the particle charge, concentration of electrons in a medium, and a certain function of velocity, but they do not depend on the particle mass. Formula (1.1) shows that as the particle energy grows, specific losses for ionization decrease at first very rapidly (inversely proportional to energy), but do this more and more slowly as the particle velocity comes closer and closer to the light velocity. However, starting from a certain large enough energy of a particle, energy losses increase on account of relativistic effects. Formula (1.1) is not quite exact: in the case of a large velocity of a particle it is necessary to take into account the so-called density effect [188], which leads to decrease of specific energy losses. Due to this fact, relativistic growth of specific losses stops and the curve becomes flat. The density effect manifests itself in condensed media earlier than in rarefied ones.
ABSORPTION OF NEUTRAL HEAVY PARTICLES IN A SUBSTANCE Deceleration of neutral heavy particles (neutrons are the most important ones) in a condensed medium occurs mostly because of direct collisions with atomic nuclei of target substance. Nuclear forces play the main role in this process. It is very difficult to obtain analytical functions in this case and one has to use either experimental data or semi-empirical functions [20]7. We should note here that energy losses by heavy neutral particles are much smaller than the corresponding losses by heavy charged particles and consequently, their depth of penetration into a medium is much larger than that of heavy charged particles.
7
Another way is to use a complex computer simulation of processes of neutron transfer in a substance using the Monte Carlo method for example or other ones. Only in some cases is it possible to obtain analytical functions of a rather complex form. Some information on these functions characterizing absorption of fast neutrons in a substance can be found in a paper by Han S. Uhm, J. Appl. Phys., 1992, v. 72 (7), p. 2549 – 2555.
PENETRATING RADIATION: GENERAL INFORMATION
16
ABSORPTION OF ELECTRONS IN A SUBSTANCE Energy losses in a substance for electrons with relatively low energy (smaller than the so-called critical energy; see below) are caused by ionization and excitation of bound electrons of a retarding medium as in the case of heavy charged particles. If electron energy is larger than the average energy of atom excitation, the formula for calculation of ionization losses by such electrons does not differ too much from formula (1.1) and can be written down in the next form [188]:
−
dE 2πe 4 mV 2 E = − (2 1 − β 2 − 1 + β 2 ) ln 2 + NZ ln dx mV 2 2 I 2 (1 − β 2 ) (1.2) 1 (1 − β 2 ) + (1 − 1 − β 2 ) 2 , 8
where E is the energy of an incident electron (total energy minus rest energy). The difference of formula (1.1) to formula (1.2) is caused by the difference between the equivalent masses of an electron and a heavy particle or two electrons and also quantum-mechanical indiscriminability of two electrons participating in the process of collision. If electron energy is lower than the average excitation energy of atoms, it is difficult to describe the process theoretically as in the case of small energy of heavy charged particles, when they capture electrons of a medium, and one has to use experimental data [244]. In the case of high-energy electrons, we have to take into account the density effect as in the case of heavy charged particles. This effect reduces ionization losses as compared to those given by formula (1.2). If the energy is very large, electrons start to lose energy effectively since the role of bremsstrahlung increases. According to the classical electromagnetic theory, a charge under 2 2 3 acceleration a emits energy (2/3)(e a /c ) per time unit. An electron can be accelerated in the electric field of a nucleus since its mass is small and acceleration is proportional to the charge of a nucleus Z divided by the electron mass m. The bremsstrahlung has the major effect upon energy losses by fast electrons. Radiation does not play an essential role in the process of deceleration of fast heavy particles. This is caused by the fact that acceleration is proportional to 1/M, and as a result, radiation gets 2 reduced (m/M) times as against radiation by electrons. We can write down the next expression for energy losses for radiation by a relativistic electron [164, 188]:
RADIATION ACOUSTICS
1 dE 1 =− , E dx x0
17
(1.3)
where x0 is the so-called radiation length different for different substances. Comparison of expressions (1.2) and (1.3) shows that energy losses for radiation increase with energy growth almost linearly, while energy losses for ionization increase only logarithmically. Therefore, in the case of large energies, losses for radiation are dominant and as the energy of electrons decreases, and ionization becomes more and more important until at some critical energy Ecr losses for ionization and losses for radiation will become 2 comparable. We can give an approximate formula Ecr ≈ 1600 mu / Z and obtain an equation for the ratio of energy losses for radiation to losses for ionization [188], (dE / dx) rad EZ . = (dE / dx) ion 1600mu 2 It follows from here that in the case of water for example, Ecr ≈ 100 MeV and in the case of lead Ecr ≈ 10 MeV. In the case of electrons with energy higher than critical, as the distance increases, the energy losses are described (on the average) by an exponential law corresponding to expression (1.3). In the case of water and air x0 is equal approximately to 36 g/cm2, for aluminum it is equal to 24 g/cm2, and for lead it is about 6 g/cm2.
ABSORPTION OF ELECTROMAGNETIC RADIATION IN A SUBSTANCE In the process of transmission through a substance electromagnetic radiation is subjected to characteristic exponential absorption in contrast to the laws characterizing absorption of heavy charged particles or electrons. The reason for this is the fact that in the process of absorption each gammaquantum is taken away from an incident beam as a result of a single act. In this case the beam intensity at the absorber thickness x has the next form [188]: I ( x) = I 0 exp(− µx) ,
(1.4)
where I0 is the intensity of incident beam and µ is the coefficient of radiation absorption. The main processes determining absorption of gamma-
18
PENETRATING RADIATION: GENERAL INFORMATION
quanta in a substance are the photoeffect, Compton scattering, and formation of electron-positron couples. In the case of gamma-quanta (γquanta) of low energy the most essential role is played by the photoeffect; the Compton absorption is dominant in the range of intermediate values of energy; and formation of couples is most important at large values of energy.
FORMATION OF NUCLEAR-ELECTROMAGNETIC CASCADES IN A SUBSTANCE Nuclear-electromagnetic cascades (cascade showers as they are called frequently or just showers) are formed in a substance in the process of absorption of penetrating radiation with quanta of very high energy sufficient for formation of multiple secondary particles. Their rise occurs as a result of a large number of single interactions, and therefore, it is not an elementary process [119, 163, 174]. In the case of large values of energy of an initial particle, the energy distribution in a cascade is almost independent of the type of the initial particle since development of a cascade is determined basically by an electron−photon avalanche. Quantitative characteristics of electron−photon cascades are studied by the electromagnetic cascade theory. The major task of this theory is determination of the function describing the distribution of particles at a certain depth in a substance with atomic number Z with respect to energy, angles, and distances from the shower axis, which is determined as the line of motion of the initial particle. A spectrum of electrons in a shower or socalled cascade curves of a very complex shape can be obtained from the solutions of the kinetic equations of the cascade theory. About 80% of electrons in the maximum of a shower have energy smaller than the critical one, and 50% of them have energy smaller than 1/3 of the critical energy. The distance of recession of an electron from the shower axis is determined basically by Coulomb scattering at the path of the order of magnitude of the radiation length. It is inversely proportional to the electron energy. In other words, in the tail of a cascade, where the energy of electrons is small, they can go away farther from the cascade axis.
EXTENSIVE AIR SHOWERS Studies of cosmic rays lead to the discovery of nuclear-electromagnetic showers in the atmosphere, i.e., showers of particles of cosmic rays covering large areas. These showers got the name of extensive air showers. They rise as a result of interaction of high-energy particles belonging to the
RADIATION ACOUSTICS
19
so-called hard component of cosmic rays with a substance in the atmosphere. Let us present basic results concerning absorption of penetrating 10 radiation in a substance. If particle energy is very large (about 10 eV and higher), the rise of cascade showers occurs in a medium (in this case it is necessary that neutrinos and muons interact with a substance in the process of some nuclear reaction). If particle energy is smaller and showers do not arise, absorption of penetrating radiation in a substance depends on the type of particles and energy. Thus, absorption of gamma-quanta (and optical radiation) occurs according to an exponential law in the whole range of these energies. Absorption of charged heavy particles (and muons) is governed by law (1.1) up to the energies when a particle captures an electron or its energy becomes equal to the average energy of atom excitation. Further absorption is determined by more complex functions, which can be obtained in a simpler way from experimental data. As a proton continues to move in a substance, its losses grow until it captures an electron of a substance atom and then losses begin to decrease. A curve characterizing dependence of losses of a proton on the so-called residual path displays a peak. This peak got the name of the Bragg peak [188]. Absorption of relativistic electrons occurs at first according to an exponential law according to formula (1.3) until the critical energy is attained. Then its absorption is determined by formula (1.2) up to the energies when the velocity of incident electrons equals velocities of electrons of atomic shells. Further absorption is determined by complex functions obtained from experimental data. We should note that the energy range, when expressions (1.2) and (1.3) become invalid and absorption of heavy charged particles and electrons is determined by functions obtained from experiments, is usually small and we can consider a particle to be totally absorbed when it attains such small energy. The grounds for theoretical treatment of the thermoradiation mechanism of sound generation are the solutions of an inhomogeneous wave equation. The right-hand side of this equation contains a function describing the power density of thermal sources of sound, which are created by absorption of penetrating radiation in a substance. In the case of fluid media, this equation has the form
∆p −
1 ∂2 p c
or
2
∂t
2
=−
α ∂Q , C p ∂t
(1.5)
20
PENETRATING RADIATION: GENERAL INFORMATION
∂2 p ∂t 2
− c 2 ∆p = Γ
∂Q , ∂t
where p is the sound pressure, c is the velocity of sound, α is the coefficient of cubical thermal expansion, Cp is the specific heat capacity of a liquid, Q is the function characterizing energy evolution (absorption) of penetrating 2 radiation or the power density of thermal sources of sound, and Γ ≡ c α / Cp is the Grueneisen coefficient. In the case of an isotropic solid, the equation has the form 1 ∂2 (3 − 4 / n 2 ) β divF ∆ − u = Qdt − div , ∫ 2 2 2 Cε ρ c ∂ t c ρ l l
(1.6)
2 ∆ − 1 ∂ rot u = − rotF . ct2 ∂t 2 ct2 ρ
(1.7)
Here cl and ct are the velocities of longitudinal and transverse waves, respectively, n = cl / ct , u is the vector of displacement of a particle of a solid, ρ is the density, β is the coefficient of linear thermal expansion, Cε is the specific heat capacity of a solid, and F is the external non-thermoelastic force applied to the unit of solid volume (the nature of this force will be explained later); F ≡ 0 if we consider only the thermoradiation mechanism of sound generation, and therefore, only thermal sources of sound are taken into account. One can see from the equations given below that the amplitude of sound pressure is proportional to the Grueneisen coefficient Γ and depends on the function of energy release Q (the power density of thermal sources of sound). The form of the function Q for different types of penetrating radiation differs. In the case of electromagnetic radiation Q( x1 , x 2 , x3 , t ) = µI ( x1 , x 2 , t ) exp(− µx3 ) ,
(1.8)
where x3 is the coordinate in the direction of propagation of radiation beam, µ is the coefficient of radiation absorption in a medium or the inverse path of radiation quanta in a medium. For example, in the case of light with the wavelength λl = 1.06 µm (radiation of a neodymium glass laser) and pure −1 water µ = 0.17 cm and in the case of CO2-laser radiation (λl = 10.6 µm) −1 and pure water µ = 800 cm . Formula (1.8) is true for electromagnetic radiation of various types, i.e., light, X-rays, synchrotron radiation, beams of γ-quanta, etc., as well as
RADIATION ACOUSTICS
21
relativistic electrons, where energy losses in the process of interaction with a substance are connected with emission of photons (radiation losses). We can write down an expression for the intensity of penetrating radiation in a beam: I ( x1 , x 2 , t ) = n p ( x1 , x 2 , t ) E ,
(1.9)
where np(x1,x2,t) is the density of particles. We have for the function of energy release Q( x1 , x 2 , t ) = n p ( x1 , x 2 , t )
dE ( x3 ) . dx3
(1.10)
One can see readily from expressions (1.3), (1.4), (1.9), and (1.10) that −1 formula (1.8) is true for relativistic electrons if we take µ = x0 . If the energy of electrons (non-relativistic) is small but larger than the average energy of excitation of substance atoms interacting with the electron beam, losses are determined by ionization. As we have mentioned above already, radiation losses increase with the growth of energy almost linearly and the losses of energy Ee increase only logarithmically. Therefore in the case of large energy Ee (relativistic velocities), radiation losses prevail and as the energy of electrons decreases, ionization plays a more and more important role until losses for ionization and losses for radiation become equal at a certain critical energy Ee,cr. For example in the case of water, Ee,cr = 100 MeV and for lead Ee,cr = 10 MeV. In the case of energies higher than the critical one, energy losses are described (on the average) by an exponential law according to formulae (1.3) and (1.8). Thus, a beam of relativistic electrons with energy of 1 GeV loses up to 90% of its energy in water and up to 99% of it in lead in compliance with exponential law (1.8), and only about 10% and 1%, respectively are lost due to ionization. In the case of heavy charged particles (protons, nuclei of various atoms, and particles in general, which take part in strong coupling), formula (1.8) is inapplicable and we can use expressions (1.1) and (1.10) to determine the form of the function of energy evolution. In the process of absorption of penetrating radiation with particles of very high energy sufficient for production of multiple secondary particles, when nuclear-electron-photon or nuclear-electromagnetic cascades rise in a substance energy absorption in a cascade can be considered as approximately exponential.
22
PENETRATING RADIATION: GENERAL INFORMATION
As for deceleration of neutral charged particles (the most important of them are neutrons) in a condensed medium, this deceleration occurs due to direct collisions with nuclei of atoms of the substance as we have mentioned above. Derivation of universal analytical functions in these cases is difficult as a rule, and one has to be content with specific calculations or empirical functions based upon experimental data. Thus, determination of the analytical form of the function of energy release, Q can be difficult. Therefore, the exponential law of absorption of penetrating radiation in a substance, and the Gaussian intensity distribution in a beam with respect to its radius, are adopted frequently when specific problems of thermoradiation generation of sound are treated. The basic conclusions of the theory also stay true in the majority of cases for other kinds of radiation, when absorption in a substance is not governed by an exponential law [156].
CHAPTER 2
Basic Mechanisms of Sound Generation by Penetrating Radiation in Condensed Media Mechanisms of sound generation are physical phenomena resulting in conversion of energy of penetrating radiation into energy of sound waves in the process of radiation absorption in a medium. Mechanisms of sound generation may depend on the type and intensity of penetrating radiation, target substance where this radiation is absorbed, and the mode of energy release. Physical phenomena of conversion of energy of penetrating radiation into sound energy (mechanisms of sound generation) are multiple and unequal in their efficiency. Here we treat only those of them which are apparently most important in the sense that they alone determine sound fields in the majority of real situations.
1. MECHANISMS CONNECTED WITH HEAT RELEASE One of the most universal physical phenomena taking place in the process of absorption of penetrating radiation in a substance is heat release. Released thermal energy can be converted partially into the energy of sound waves in different ways. 23
24
BASIC MECHANISMS OF SOUND GENERATION
In the case of moderate density of energy released in a medium when phase transitions do not occur, the major contribution into the process of sound generation belongs usually to thermal cubical expansion. This is a socalled thermal or thermoradiation mechanism of sound generation. Sometimes it is called a thermoelastic mechanism, when sound excitation by penetrating radiation in a solid is discussed. A distinctive feature of the thermoradiation mechanism is the fact that the basic fundamental features of sound fields can be described within the framework of a linear model corresponding to the acoustic approximation in the case of calculation of hydrodynamic or elastic disturbances. It has provided an opportunity to develop a very effective theory of sound generation in a liquid (e.g., by laser radiation), which agrees well with experimental data (see reviews by Bunkin and Komissarov [41], Lyamshev [128, 129], and Lyamshev and Sedov [142]). Recently a theory of thermoradiation sound generation by penetrating radiation has been developed [156]. The pattern of sound generation in the case of large intensities of radiation (and more precisely, in the case of large densities of energy delivered into a medium) looks more complex. The effects developing in this case are of nonlinear character, and the design of an exhaustive theory of these phenomena is really far from being finished. Some results concerning sound generation in a liquid by high-intensity laser radiation can be found in reviews by Lyamshev [128] and Lyamshev and Naugol’nykh [141]. Nonlinear effects similar to the ones characteristic to interaction of powerful laser radiation with a substance also arise in a medium in the process of absorption of penetrating radiation in it, when the density of energy released in a medium is sufficient for phase transitions [86]. An exhaustive theory of such phenomena has been developed to an even less extent than in the case of absorption of laser radiation. Further we consider only some examples of such phenomena. Let us treat in more detail the basic mechanisms of sound generation by penetrating radiation.
2. THERMORADIATION GENERATION OF SOUND Many studies have been devoted to the thermoradiation mechanism of sound generation by various types of penetrating radiation [8, 9, 11, 16, 23, 28, 36, 54 – 56, 58, 68, 70, 81 – 83, 85, 103, 104, 151, 153, 154, 158, 159, 193, 194, 199, 200, 209, 212, 224, 225, 231, 235, 242, 244, 255, 261]. Sets of hydrodynamic equations or equations of dynamic theory of thermoelasticity and equations of the field of radiation of a given type are initially required for derivation of the equation of thermodynamic
RADIATION ACOUSTICS
25
generation of sound in a liquid or isotropic solid by penetrating radiation. Linearizing a set of equations and ignoring the effects of thermal conductivity and viscosity in the case of a liquid, we write down an inhomogeneous wave equation of sound generation in the form [259],
∆p −
1 ∂2 p c
2
∂t
2
=−
α ∂Q . C p ∂t
(2.1)
In equation (2.1), p is the sound pressure, α is the coefficient of cubical thermal expansion, Cp is the specific heat capacity of a liquid, Q is the function characterizing the power density of thermal sound sources arising due to absorption of penetrating radiation in a liquid, and c is the sound velocity in a liquid. It follows from this equation that in the case of a sound field of high frequency or sound pulse of small duration, the amplitude of a signal is proportional to the quantity αc2/Cp or the Grueneisen parameter (coefficient) Γ. The same may be said also about sound stress in a solid. The power density of thermal sources of sound depends on the type of penetrating radiation and the target material. Many papers studied the dependence of acoustic signal amplitude on the energy of penetrating radiation and the substance and geometry of the target. Thus, Borshkovskii and Volovik [36] studied the dependence of the amplitude of acoustic signal excited in thin metal (aluminum) plates by pulsed beams of electrons and protons. Some results are given in Fig. 2.1, where dependences of acoustic signal amplitude on energy of electrons and protons are presented. As the authors note, these results are the evidence of the fact that the main contribution to acoustic vibrations from electrons and protons flying through thin metal plates derives from energy losses for ionization.
Figure 2.1 Dependences of acoustic signal amplitude on the energy of (1) protons and (2) electrons.
26
BASIC MECHANISMS OF SOUND GENERATION
Volovik and Ivanov [53] demonstrated that the ratio of amplitudes of acoustic signals excited by pulsed beams of protons and electrons in aluminum and piezoelectric ceramics is equal to the ratio of the corresponding Grueneisen coefficients (if the density of absorbed energy is the same). This was considered by the authors as the proof of the thermoelastic nature of the signal in these substances. Analogous estimations of acoustic signal amplitude (or experimental data) were made in the majority of other relatively early papers [28, 242, 261]. The authors of later papers studied the shape of a signal together with its amplitude. Blazhevich et al. [28] used the shape of an acoustic pulse excited in a solid sample by a pulse of electronic beam for determination of density of energy absorbed in the sample. In this case the shape of a one-dimensional acoustic pulse is connected with the density of energy absorbed in a sample E(x) by the relationship, v ( x, t ) ≡ −
Γ E ( x − ct ) , 2
(2.2)
where c is the sound velocity in a sample. Lyamshev and Chelnokov [152, 153, 158, 159] studied theoretically sound excitation by penetrating radiation in a liquid and solid in an approximation of the thermal mechanism. Many papers were devoted to investigation of the problem of the mechanism of generation of acoustic vibrations by penetrating radiation in a liquid [16, 68, 235, 255]. For example, Golubnichii, Kalyuzhnyi, and Korchikov [68] established that the rise of the acoustic signal produced by an absorbed electron beam in a liquid is determined by the thermal mechanism of sound generation. Performing accurate measurements, they demonstrated that the measured amplitude of acoustic signal in a liquid coincided with the one predicted theoretically and in the case of changing of water temperature around 4°C, the acoustic signal changed its polarity which was explained by the change of the sign of the absorption coefficient of water. Analogous results were obtained by Balitskii et al. [16] who studied, however, an acoustic field generated by a totally absorbed electron beam in water, and not by a beam, which was partially absorbed in a cell with water and partially transmitted through it [68]. Similar results demonstrating that, in the case of protons, the thermal mechanism of sound generation in water is dominant, were obtained by Danil’chenko et al., Levi et al., and Sulac et al. [85, 235, 254, 255]. Many papers devoted to acoustic detection of single high-energy particles (the DUMAND Project) analyze the parameters of the sound field
RADIATION ACOUSTICS
27
in the near-field zone of a nuclear-electron cascade arising in water [11, 56, 194, 200, 214]. Thus, using equation (2.1) and calculating in different ways the function Q(x, y, z, t), we can obtain the shape and amplitude of a produced acoustic pulse. The time dependence of the function Q is always taken in the form of the delta-function δ(t) since the time of the cascade rise is much smaller than other characteristic times of the process of sound generation, and the spatial dependence is determined on the basis of various direct calculations of density of energy release in the cascade taking into account possible approximations. For example, Volovik et al. [58] give the next expression for the amplitude of acoustic signal in sea water:
Pmax =
0.44ϕ (r ) E 18 Pa , r 10
where E is the energy of a particle (in eV) and r is the distance from the cascade axis (in meters). The factor ϕ(r) allows for deviation from the law p(r) ~ r−1/2, because of absorption or sound refraction for example and it is taken approximately equal to one. For example in the case of a particle with the energy E = 1017 eV, the amplitude of acoustic signal at a distance 100 m from the cascade axis is about 4.10−3 Pa. Figure 2.2 presents shapes of acoustic pulses for various dependences of the function of power density of sound sources Q [58].
Figure 2.2 Shapes of acoustic pulses for various dependences of the function of density of energy release Q. (1) The case of distribution of energy released in an electromagnetic cascade according to the Nishimura-Kamata-Greisen model; (2) the case of uniform distribution in a cylindrical region of the length of 5 m and radius of 2 cm; (3) the area with the Gaussian distribution of density of energy release.
28
BASIC MECHANISMS OF SOUND GENERATION
3. INITIATION OF MICROSHOCK WAVES Passage of penetrating radiation through a substance may cause microshock waves. Thus, for example in the case of passage of fission fragments through a liquid, shock waves may arise in it along the track of the fragments [72, 233]. On the other hand, delta-electrons arising in the process of passage of ionizing particles through a liquid produce overheated micro-regions (thermal sources) in it. Explosive expansion of these regions gives birth to a shock wave [4]. The rise of shock waves is an essentially nonlinear effect. The theory of such processes is also far from being completed. Following Anoshin [4], let us perform estimation of parameters of shock waves produced by delta-electrons. In order to produce micro-regions capable of explosive expansion, it is necessary for the energy lost by an ionizing particle to be localized initially in a small enough volume. This condition may be satisfied in the case of delta-electrons knocked out by an ionizing projectile particle. The path l(Eδ) (in cm) of a delta-electron depends on its energy Eδ (in eV) in the following way: l ( Eδ ) = 0.58 ⋅ 10 −12 ⋅ AEδ2
1 , ρ0Z
(2.3)
where ρ0 is the medium density and Z/A is the ratio of the number of electrons to the molecular mass. As the energy of a delta-electron decreases, its trajectory differs more and more from a straight line and becomes similar to a coil. For example, already at Eδ = 10 keV the average path of an electron is a half of the path l(Eδ), i.e., 0.5l(Eδ). The number ν of water molecules per the path l(Eδ) is determined as
ν=
1/ 3
l ( Eδ ) l ( Eδ ) 4πρ 0 N A = d 2 3A
,
(2.4)
where d is the diameter of the spherical volume V1 occupied by a single water molecule and NA is the Avogadro constant. If the energy of a deltaelectron is small enough, its track is located within the sphere of the volume V0, which is determined according to the formula V0 = νV1 =
l ( Eδ ) A . ρ0 N Ad
RADIATION ACOUSTICS
29
The energy transmitted on average to a single molecule is equal to ∆E = Eδ /ν = Eδ d / l(Eδ). If we choose (proceeding from reasonable assumptions) for water ∆E = 30 eV when micro-explosive production of bubble nuclei occurs, the parameters making possible explosive expansion of overheated regions in water have the values Eδ = 1.23 keV, ν ≈ 41, and 3 V0 = 1.23⋅10−21 cm . The radius of the sphere of the volume V0 is equal to −8 a0 = 6.64⋅10 cm. The density of energy release Q in the thermal peak, which is determined according to a formula Q = Eδ/V0 = EδAd/[l(Eδ)ρ0N0], in this case is equal to 16.09⋅104 J/cm3. Comparing it to the caloricity of a common explosive, one can see that the latter is equal to 4.19⋅103 J/cm3 approximately. According to the theory of underwater explosions, pressure in a shock wave psh(r,t) can be written down as (2.5)
psh (r , t ) = psh max (r ) exp(−t / Θ) ,
where psh max(r) is the maximum pressure (in Pa), r is the distance from the centre of the spherical volume of thermal peak (in cm), t is time, and Θ is the time constant (in seconds). In this case it is possible to demonstrate that psh max (r ) = 1.37 ⋅ 10 2 r −1 Pa , Θ(r ) = 1.07 ⋅ 10 −10 (7.18 + log r ) at r > 102 a0. The spectrum of the pulse psh(r,t) has the form
[
S (ω ) = psh max (r ) Θ − 2 (r ) + ω 2
]
−1 / 2
.
(2.6)
If frequencies ω < 1 / Θ(r), the spectrum is uniform and can be written down as S (ω ) ≈ psh max (r ) ⋅ Θ(r ) .
Root-mean-square pressure at the distance r from a single thermal peak in water is p = 7.76 ⋅ 10 −8 (7.18 + log r ) ⋅ (1 / r ) Pa/Hz1/2 .
30
BASIC MECHANISMS OF SOUND GENERATION
The number of such thermal peaks per unit length of a track of a relativistic electron is equal to the number of delta-electrons with Eδ = 1.23 keV (according to Anoshin [4], it is equal to 17.32 cm−1). It is interesting to note that it is possible to obtain the following estimate for the intensity I (in W/cm2) of acoustic radiation in water from a single high-energy particle, which produces in the process of its entry to the atmosphere an extensive air shower consisting mostly of electrons [4]. I = 10 −29 E0 / R ,
(2.7)
where E0 is the energy of a particle (in eV) and R is the distance from the axis of an extensive air shower (in cm). Apparently, now it is possible to 17 detect extensive air showers at E0 ≥ 10 eV according to underwater acoustic radiation. When a cosmic particle of super-high energy gives birth to a shower of secondary particles directly in water, the density of radiating centers is much larger than in the case of an extensive air shower in the atmosphere. This provides an opportunity to lower the threshold of detection with respect to particle energy. Apparently, this takes place in mountain lakes, for example. The considered opportunity for acoustic detection of a high-energy cosmic particle does not concern apparently muons and neutrinos because of their large penetrating power. These particles produce cascades deep in the ocean and Earth and their sound fields are generated due to the thermal mechanism of sound excitation. It is interesting to note that the estimated value of intensity of sound field of an extensive air shower does not contradict the hypothesis by Lyamshev et al. [150] on generation of noise in the calm ocean by aggregate cosmic radiation.
4. BUBBLE MECHANISM Many authors discuss the possibility of sound generation in a liquid by penetrating radiation on account of the rise, oscillation, and collapse of microscopic bubbles at tracks of particles constituting a given type of ionizing radiation1. The theory of the bubble mechanism of sound generation is far from being completed. This is connected with the fact that a satisfactory description of arising phenomena needs to take into account the complex processes of nonlinear dynamics of a single bubble and a set of bubbles, and nonlinear effects in a liquid. For example, rough estimations 1
See [17, 57, 62, 64, 65, 150, 217, 238].
RADIATION ACOUSTICS
31
are conducted within the framework of a linear theory. A bubble arising as the result of interaction of a particle with a liquid is treated as a spherically symmetric source of sound with efficiency connected with the value of Q(x, y, z, t) in the place of “absorption” of the particle. The total effect from microscopic bubbles of a certain kind depends on both the relative amount of such microscopic bubbles and the efficiency of ultrasonic generation by a single bubble. Estimations demonstrate [57] that an essential contribution to sound radiation is made by quasistable microscopic bubbles and not only collapsing bubbles of a size of about 10−7 – 10−6 cm. In this case the acoustic signal imitates a signal from the thermal mechanism of sound generation (but it is much stronger). Recent experimental papers are evidence of the fact that the bubble mechanism of sound generation may be realized apparently (under normal experimental conditions, i.e., at atmospheric pressure and room temperature and in stable liquids) if penetrating radiation consists of heavy particles, e.g., fragments of fission nuclei [17, 71]. Further studies must clear up this question.
5. THE CHERENKOV MECHANISM When a particle moves in a medium with a velocity exceeding the phase velocity of wave propagation in this medium, it emits waves which are called the Cherenkov radiation. The intensity of the radiation and its characteristics (dispersion and polarization) depend on the nature of the waves and particle properties. The Cherenkov radiation of sound by a particle moving with a supersonic velocity in a solid can be described using a phenomenological model if a force (per unit volume) acting on a solid (lattice) and caused by a particle is introduced into the equation of the dynamic theory of elasticity [101]. The equation for the longitudinal component of displacement vector u takes the form, 1 ∂2 D div u = ∆ − ∆δ (r − vt ) , 2 2 2 c t c ∂ ρ l l
(2.8)
where D is the dimensional constant coinciding in order of magnitude with the bond energy of a particle with the lattice, v is the velocity of a particle, and cl is the velocity of longitudinal waves. The transverse component of the displacement vector is identically equal to zero since the acting force is
32
BASIC MECHANISMS OF SOUND GENERATION
“longitudinal”. In this case the particle emits longitudinal sound waves and the spectral density of radiation energy Iω is proportional to the third power of sound frequency ω (see [101]:
Iω =
D2 4πρcl4
ω3 .
(2.9)
In the case of motion in a metal, a charged particle produces an electromagnetic field around itself. This field disturbs the equilibrium of conductivity electrons. The latter move ions due to the bond with the lattice. If the particle velocity is larger than the sound velocity, this mechanism leads to generation of not only longitudinal waves but also transverse sound waves [102]. Spectral densities of radiation of longitudinal Il(ω) and transverse It(ω) sound waves depend essentially on the properties of a particle and metal (the velocity of a particle, free path of electrons in a metal, Debye frequency, Fermi energy, etc.). In the case of low frequency, It(ω) ≥ Il(ω). At a certain frequency, ω = ϖ the values of It(ω) and Il(ω) become equal and at ω >> ϖ we have Il(ω) >> It(ω). The frequency ϖ depends on the particle velocity v and the free path of an electron l. The larger l and v are, the larger the value of ϖ is. The value of ϖ attains its maximum in the case of ultrarelativistic particles (v ≈ u) at l > δ0(vF cl)1/2, where vF is the Fermi velocity of conductivity electrons of a metal (~ 108 cm/s), δ0 ≈ 10−5 cm, and ϖ ≈ 104 s−1. The total intensity of radiation at all frequencies of transverse sound waves is much smaller than the analogous value for longitudinal waves It ≈ ≈ It(Vcl / u2)2 > cl) being close to π/2. Deceleration of particles (finiteness of trajectory length) smears the cone to some extent. This smearing depends on frequency. It is larger, the smaller the frequency. In the case of the lowest possible frequencies (cl / ω >> L, L is the length of particle track), radiation is quite dissimilar to the Cherenkov radiation. It is determined by the average acceleration of a particle and looks like bremsstrahlung of light. The Cherenkov mechanism of sound generation contributes noticeably to the sound field emitted by a particle only at very high hypersonic frequencies (the spectral density of emitted energy is proportional to the third power of frequency). Thus, according to Borshkovskii and Volovik
RADIATION ACOUSTICS
33
[36], in the case of frequency range of hundreds of kilohertz, which may be important in practice, the ratios of acoustic energy emitted by particles due to the Cherenkov mechanism of sound generation to the acoustic energy emitted due to the dynamic (see below) and thermal mechanisms are 10−13 and 10−21. This is the evidence of relative weakness of this mechanism of sound generation in the low frequency range (in comparison with hypersound). Similar to electrodynamics, with the transition radiation connected with motion of a charged particle through media with different electromagnetic properties, the transition radiation can exist in acoustics also. This radiation is connected with the change of acoustic properties of a medium, where penetrating radiation is absorbed [171]. As in the case of the Cherenkov radiation, this radiation is quite weak. We will not discuss it in detail. There are mechanisms of sound generation specific to this or that type of radiation or target material. Let us discuss some of them briefly.
6. STRICTION MECHANISM OF SOUND GENERATION Microstiriction occurs in the field of ions in the process of medium ionization. It manifests itself noticeably in macroscopic effects [10]. A transiting charged particle (or any other particle capable of ionization) produces N1(0) pairs of ions per unit length of track in a medium. The number of these ions Nt(t) decreases sharply in time because of their recombination. Each ion attracts molecules of a medium by its field and creates local clouds. Microstrictional compression can play a significant role in the process of sound emission by charged particles in the case of a small coefficient of thermal expansion of a medium. In particular, the experimental results by Levi et al. [235] apparently can be explained with its help. It was revealed in these experiments that a sound pulse from a beam of charged particles in water vanishes and changes its sign not at T = 4°C when the coefficient of thermal expansion of water α = 0, but at T = 5.7°C when α ≈ 10−5 K−1. Amplitudes of pulses become equal at α ≈ 10−5 K−1, i.e., compensation of thermal expansion by strictional compression is possible in this case. Fast alternating striction in the field of a moving particle and in the collective field of beams also exists in the field of ions apart from microelectrostriction. Let us consider the strictional mechanism of sound generation in terms of laser excitation of sound. The equation of sound generation in the process of action of laser radiation on a liquid medium,
34
BASIC MECHANISMS OF SOUND GENERATION
which takes into account thermal and strictional mechanisms, takes on the following form [43]: ∆p −
1 ∂2 p c
2
∂t
2
=−
α ∂Q 1 + C p ∂t 8π
∂ε ρ ∆ E 2 ; ∂ρ T
(2.10)
here ε is the dielectric constant of a liquid, E is the strength of electric field of laser (optical) radiation in a liquid, the angular brackets mean averaging with respect to the period of optical oscillations, (∂ε /∂ρ)T is the derivative of dielectric constant with respect to density in the case of constant temperature. The second term in the first part of equation (2.10) corresponds to the strictional mechanism of sound generation. Let us estimate the order of magnitude of the expression ∆〈E2〉. Monochromatic sound oscillations with the frequency ω can be excited due to the strictional mechanism in the case of two laser beams crossed under the angle θ and with frequencies ω1 and ω2, ω1 - ω2 = ω, and 2(ω1,2/u)n sin(θ/2) ~ ω / c = k. We have 1 8π 1 + (ak ) 2 + (aµ ) 2 q0 , ∆ E 2 ~ + k 2 + µ 2 E02 ~ nu a2 a2
(2.11)
where n = u1/2 is the refraction coefficient of a medium, k is the wave number, a is the radius of radiation beam, µ is the coefficient of radiation absorption in a liquid, E0 is the amplitude of the strength of electric field, u is the light velocity, c is the sound velocity in a liquid, and q0 is the intensity of light pulse in the middle of a beam. Since |∂Q / ∂ t| ~ µq0ω, the ratio of the first term in the right-hand side of equation (2.10) (corresponding to the thermal mechanism of sound generation) to the second term is equal to
αanuω µα . C p ( ρ∂ε / ∂ρ )T 1 + (ak ) 2 + (aµ ) 2 This relationship shows that the strictional mechanism of sound generation prevails over the thermal one only in the range of very high or very low frequency. In the case of water for example, the thermal mechanism is dominant with respect to the strictional mechanism at µ ≥ 0.2 cm−1 in the frequency range from 102 to 109 Hz.
RADIATION ACOUSTICS
35
7. SOUND GENERATION IN THE PROCESS OF PULSED RADIOLYSIS Pulsed radiolysis is the mechanism of release of latent energy on account of substance decomposition under the effect of pulses of penetrating radiation [52, 59 – 61]. In particular, slow electrons arising in the process of substance ionization can be absorbed effectively by some halogencontaining liquids, i.e., reactions of the following type may occur: ABC n + e − = ABC n −1 + C − , where ABCn is the formula of a molecule of a liquid containing a certain halogen C in it. Further, this released energy may be transformed into sound energy with the help of the mechanisms of sound generation already considered above. These may be, for example, the thermal or bubble mechanisms of sound generation. In this case, if the density of released latent energy of a substance exceeds the density of energy evolved on account of absorption of penetrating radiation, the determining contribution to acoustic field in such radiation-unstable substance belongs to the effect of pulsed radiolysis [60].
8. DYNAMIC MECHANISM If penetrating radiation affects a substance, transfer of momentum from radiation quanta to atoms of a medium occurs. This phenomenon accompanied by excitation of sound waves is called the dynamic mechanism of sound generation [7, 63, 165]. In crystals the effect depends on the reciprocal directions of crystal axes and the velocities of particles constituting penetrating radiation [165]. Like the thermal mechanism of sound generation, the dynamic mechanism takes place in the case of any kind of penetrating radiation and manifests itself in both liquids and solids. In solids, however, the dynamic mechanism may be of fundamental importance because of the existence of transverse sound waves there [160]. The point is that transverse waves in solids under conditions of thermal mechanism result only from transformation of longitudinal waves at interfaces and medium inhomogeneities. As a result of action of the dynamic mechanism, longitudinal and transverse waves are excited in all cases and even in homogeneous isotropic solids.
36
BASIC MECHANISMS OF SOUND GENERATION
Dynamic stress in solids in the process of transmission of beams of charged particles through them was determined by Nasonov [165]. In the case of small energy in a beam (the energy of several hundreds of electronvolts for plates of thickness of the order of magnitude of 0.01 cm), the pressure p produced by a beam incident on a plate does not depend on the target substance, and increases linearly with energy: p = n0p0ν0, where p0, ν0, and n0 are the momentum, velocity, and density of a beam of incident particles. In the case of large energy of a beam, the dependence of pressure on energy is of more complex character. Results of numerical calculation of dynamic pressure conducted by Nasonov [165] for aluminum, copper, and lead plates of thickness of 0.01 cm at different values of energy, are given in Figure 2.3.
Figure 2.3 Dynamic pressure upon a plate vs. energy of a beam in a plate. Broken curves correspond to proton beams and solid curves correspond to electron beams. (1) Copper, (2) lead, and (3) aluminum.
Sound generation under the dynamic mechanism can be treated within the framework of linear approximation. In this case, equations of sound generation in a solid have the form [160], 1 ∂2 div F ∆ − div u = − , 2 2 cl ∂t cl2 ρ (2.12) 2 ∆ − 1 ∂ rot u = − rot F , ct2 ∂t 2 ct2 ρ
RADIATION ACOUSTICS
37
where u is the displacement, cl and ct are the velocities of longitudinal and transverse sound waves, ρ is the medium density, and F is the dynamic force applied to unit volume. Comparing equations (2.1) and (2.12), one can see that the dynamic mechanism of sound generation produces a source of both longitudinal and transverse waves in a solid as distinct from the thermal mechanism, which excites sources of longitudinal waves only. In the case of penetrating radiation with quanta consisting of ultrarelativistic particles (e.g., photons), the dynamic force is equal to F = Q/u, where u is the light velocity. In order to compare the efficiency of sound excitation by the thermal and dynamic mechanisms, it is necessary to rewrite expressions (2.1) in terms of the displacement vector u and add dynamic sources to the right-hand side:
α 1 ∂2 div F ∆ − ∆Qdt − div u = . ∫ 2 2 2 Cpρ c ∂ t c ρ l l
(2.13)
One can see readily from here that under the condition F = Q/u and for the majority of substances, the ratio of displacements in sound waves excited due to the dynamic mechanism (the second term) to analogous displacements caused due to the thermal mechanism (the first term) is of the order of magnitude of the ratio of the velocity of longitudinal sound wave to the light velocity: cl/c ≅ 10−5. Thus, accounting for the dynamic mechanism of sound generation for ideal liquids (where only longitudinal waves can propagate) gives us only an insignificant correction to the sound field produced by the thermal mechanism of sound generation. Only hypersonic frequencies may be the exception. These frequencies are excited by the thermal mechanism inefficiently because of thermal conductivity of a substance, which is not taken into account in expression (2.1). However, in the case of solids, when the field of transverse waves, which cannot arise because of reflection or scattering of longitudinal waves, is studied, the dynamic mechanism of sound generation together with the thermal mechanism may provide a significant contribution.
38
BASIC MECHANISMS OF SOUND GENERATION
9. OTHER MECHANISMS OF SOUND GENERATION As for other mechanisms of sound generation by penetrating radiation, we should note the mechanism of excitation of elastic waves with the help of the inverse piezoelectric effect [53, 122] and the transition mechanism already mentioned above. The first mechanism is connected with mechanical deformation of piezoelectric ceramics under the action of an electric field of penetrating radiation. We have to mention also that if weakly ionized plasma is produced in the area of absorption of penetrating radiation, interaction of electric and magnetic fields with plasma may violate the stable state of a medium and therefore, sound waves may be generated [123]. It is also possible that new data on as yet unknown mechanisms of sound generation may be obtained in the course of further theoretical and experimental research. Such research will provide an opportunity to use the methods of radiation acoustics for the solution of applied and basic problems even more widely. All considered mechanisms of sound generation have different restrictions in intensity, nature, and frequency range of emitted sound waves. Exact quantitative values can be obtained by the solution of specific boundary problems. It is essential in this case that in the majority of real situations all possible mechanisms of sound generation in solids contribute only small corrections to the sound field produced by the thermal mechanism of sound generation [55]. This is also true in the case of liquids if there are no phase transformations. Thus, the above discussion of the determining role of the thermal mechanism of sound generation in liquids and solids in the case of moderate density of energy released in a medium explains the importance of detailed studies of generation of sound fields in the approximation of this mechanism.
CHAPTER 3
Thermoradiation Excitation of Sound in a Homogeneous Liquid This chapter considers thermoradiation excitation of sound in a homogeneous liquid by intensity modulated penetrating radiation. Basic equations and the technique of solution of boundary problems are discussed in details. Specific features of excitation of monochromatic sound in a liquid half-space with the undisturbed surface (boundary) as well as the case when the liquid surface is characterized by large or small (in comparison with the sound wavelength) unevenness are considered. Efficiency of thermoradiation generation of sound in a liquid by penetrating radiation is discussed.
1. EQUATION OF THERMORADIATION GENERATION OF SOUND We understand the thermoradiation mechanism as a mechanism of sound excitation by penetrating radiation in a liquid, when a medium expands in the area of absorption because of heating due to radiation absorption but the aggregate state of a substance and its thermodynamic parameters do not change and the expansion velocity of the heated volume is essentially smaller than the velocity of sound propagation in the medium. This provides an opportunity to write down the set of equations of conservation 39
40
THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID
in the linearized form and derive a linear equation of thermoradiation generation of sound on its basis. Thus, an initial set of linearized equations of hydrodynamics complemented with equations of state and heat conductivity of a liquid is [43, 120]:
ρ
∂v η = −∇p ′ + η∆v + ξ + ∇(∇ ⋅ v ) , ∂t 3
∂ρ ′ ∂s ′ = − ρ (∇ ⋅ v) , ρT = κ∆T ′ + Q , ∂t ∂t (3.1) ∂p αc 2 ρT ∂p ρ ′ + s ′ ≡ c 2 ρ ′ + p ′ = s′ , Cp ∂s v ∂ρ s ∂s s′ = ∂T
Cp ∂s α T ′ − p′ . T ′ + p ′ ≡ T ρ p ∂p T
Here ρ, v, p, T, and s are the density, velocity, pressure, temperature, and specific entropy of a liquid, respectively; a stroke means a small increase of the corresponding parameter of a medium; η and ξ are the coefficients of shear and bulk viscosity; Cp and Cv are the specific heat capacities; v is the specific volume; α = v−1(∂v/∂T)p is the coefficient of cubical thermal expansion; κ and χ are the coefficients of thermal conductivity and diffusivity (χ = κv/Cp); Q is the power density of thermal sources produced due to absorption of penetrating radiation in a liquid, Q = − (∇⋅Π), and Π is the density of energy flux of penetrating radiation. We can obtain the desired equation of thermoradiation generation of sound on the basis of set (3.1): ∆p ′ −
1 ∂ 2 p′ c
2
∂t
2
+ 2Γ∆
∂p ′ αρT = Cp ∂t
4 / 3η + ξ ∂s ′ α ∂Q , − χ ∆ − ρ ∂t C p ∂t (3.2)
Γ≡
Cp 1 2 4 / 3η + ξ + χ − 1 . C C 2 ρ v
RADIATION ACOUSTICS
41
Let us turn to the physical meaning of some terms in equation (3.2). First of all, one can see that the third term in the left-hand side of the equation characterizes sound attenuation caused by viscosity and heat conductivity of a liquid. In the majority of cases, this term may be omitted in the process of solution of boundary or initial problems without any restrictions to generality. It can be taken into account separately while considering the final result as is done usually in acoustics. Terms in the right-hand side of equation (3.2) describe sound sources. It is possible to ignore the role of the first of them if we ignore the influence of viscosity and heat conductivity of a liquid. This is possible if the next conditions are satisfied: l 2 l 2 τ > l2min/ν, we have to note that the quantity (1/τν)1/2 ~ (ω/ν)1/2 characterizes the penetration depth of a so-called viscous wave and therefore, the condition τ > (ω/ν)1/2. In other words, the minimum dimensions of the area of heat evolution (thermal source) must be essentially larger than the depth of wave penetration. If the condition τ > ka2/(µL) or L >> ka2/(a/L) that is the condition of the Fresnel diffraction. Taking into account these conditions and the fact that radiation intensity rapidly decreases already at ρ = a, we may change |r′ − r| and |r′* − r| in the denominator of the integrand in expression (3.13) for L and change them in the exponential factor for L = z′ + (ρ − ρ′)2 / 2L. After this substitution and integration, the solution takes on the form [43],
p=i
ρ2 mAαc µk exp[(ik − Γ) L] . I 0 exp − a2 Cp µ2 + k2
(3.25)
Naturally, a sound beam in the near wave zone is not subjected to divergence and its width (with respect to pressure amplitude) coincides with the width of the penetrating radiation beam. Thus, we arrive at an important conclusion that while in the far wave field, where sound waves are spherical, the amplitude of sound pressure p is determined by the total power P of penetrating radiation, in the near wave field the amplitude of sound is determined by the beam intensity according to expression (3.25).
52
THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID
4. A LIQUID HALF-SPACE WITH LARGE-SCALE ROUGHNESS OF BOUNDARY Above we have considered the particular features of a sound field of a radiation-acoustic source in a liquid bounded by an even surface. In real conditions a liquid surface is uneven because of many reasons. Therefore, it is useful to estimate the effect of boundary roughness upon the sound field generated by a radiation-acoustic source. At first we consider the influence of unevennesses which are large in comparison with the sound wavelength in a liquid [108, 133]. We take random unevennesses in order to consider a general case. Let a penetrating radiation beam intensity-modulated with the sound frequency ω be incident along the axis z upon the uneven surface of the liquid half-space determined by the equation z = ξ(x, y). Thermal sound sources and a sound field arise in a liquid due to absorption of penetrating radiation in it. As we have mentioned above, they are described by the inhomogeneous reduced wave equation (∆ + k 2 ) p = i
αA mωµI ( x, y ) exp[− µ ( z − ξ ( x, y ))] . Cp
We write down the solution of this equation on the basis of the reciprocity principle (see equation (3.19)),
p ( x, y , z ) =
1 4π
iαA p x y z x y z mωµI ( x1 , y1 ) × ( , , ; , , ) 0 1 1 1 ∫ C p Ω (3.26)
exp{− µ [ z1 − ξ ( x1, y1 )]} dx1dy1dz1 , where p0(x, y, z; x1, y1, z1) is the solution of the regular problem of sound scattering at the boundary when the source of the regular field is located at the point (x, y, z), i.e., at the point, where it is necessary to determine the desired field; and Q is the volume occupied by thermal sound sources produced by the effect of penetrating radiation. Assuming that the point (x, y, z) is located in the far wave field, we can represent the incident regular wave by an expression, pi =
e ikR exp[i (αx1 + βy1 + γz1 )] , R
(3.27)
RADIATION ACOUSTICS
53
where R = (x2 + y2 + z2)1/2. Here α, β, and γ are the components of the wave vector directed from the point (x, y, z) to the coordinate origin. In the considered case of the boundary with large-scale unevenness, the solution of the diffraction problem can be written down in the Kirchhoff approximation, i.e., under the assumption that reflection at each point of the surface occurs in the same way as from the infinite plane tangential to the boundary surface at this point [18, 125]. In this case the total field at the boundary consists of the incident pi and reflected pr = Wpi waves, where W is the reflection coefficient. We may take W = −1 at the liquid−air boundary. The scattered field in a liquid half-space in the general case can be represented by the Kirchhoff integral. However, in the case under consideration, when the distance from the point (x1, y1, z1) to the boundary is not larger than the sound wavelength (just this length of the region of the thermal sound sources along the axis z provides the optimal mode of sound generation as has been mentioned before), the scattered field in the layer of the depth k−1 can be represented at the boundary in an approximate form, pr = −
e ikR exp{i (α ′x1 + β ′y1 + γ ′z1 ) − i[γz1 + γξ ( x1 , y1 )]} , R
(3.28)
α ′ = k sin θ ′ cos ϕ ′ , β ′ = k sin θ ′ sin ϕ ′ , γ ′ = k cos θ ′ , and the angles θ′ and ϕ′ are expressed with the help of the coordinate angles of the wave vector of the incident wave θ and ϕ in the following way:
ϕ ′ = arctan
(tan θ sin ϕ + 2∂ξ / ∂y )(1 − tan θ cos ϕ ⋅ 2∂ξ / ∂x) , (1 − tan θ sin ϕ ⋅ 2∂ξ / ∂y )(tan θ cos ϕ + 2∂ξ / ∂x)
θ ′ = arctan
tan θ cos ϕ + 2∂ξ / ∂x 1 + tan 2 ϕ ′ . 1 − tan θ cos ϕ ⋅ 2∂ξ / ∂x
The validity of expression (3.28) is limited by the conditions of applicability of the Kirchhoff approximation and, in particular, by the condition of the absence of multiple reflection of the incident wave at the boundary. As can be demonstrated readily, this corresponds to imposing a certain restriction upon the slip angle of the incident wave, i.e., the following condition must be satisfied: (π/2 − θ) > 3δ , where δ is the meansquare angle of inclination of boundary unevennesses.
54
THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID
As we consider only mildly sloping unevennesses, that is equivalent to the satisfaction of the condition ∂ξ/∂x 1, the integrand in expression (3.36) is essential only for the values of ρ close to zero. Then the correlation coefficient can be expanded into a series and we can take only two terms of the expansion: | N ′′(0) | ρ N (ρ ) ≈ 1 − ρ 2 0
2
,
where ρ0 is the correlation length of the boundary unevenness. Let us determine the mean-square angle of unevenness inclination by a relationship, ∂ξ tan 2 σ = ∂ρ
2
=
σ2 ρ 02
| N ′′(0) | ,
(3.39)
and obtain an expression for the characteristic function, γ 2 tan 2 δ 2 f (−γ , γ ; ρ ) ≈ exp − ρ . 2
(3.40)
Substituting expression (3.40) to formula (3.36) and assuming that tan2σ = 2(σ/ρ0)2 for the normal distribution law, we represent finally the average intensity of the sound field of thermoradiation sources in the case of largescale homogeneous isotropic unevenness of the boundary in a liquid in the form, 2 I 0αmc 1 µ 2 a 4 cos 2 θ 2 〈| p | 〉 = × 2C p R [( µ / k ) 2 + cos 2 θ ] 2 1 + ∆2 cos 2 θ
(3.41) ρ 2 ∆2 sin 2 θ , exp − 0 σ 4( ∆2 cos2 θ + 1)
RADIATION ACOUSTICS
59
where ∆ = 21/2kaσ / ρ0 is the dimensionless parameter. We have to note that the average field intensity can be also obtained readily for the case of anisotropic unevenness. It is necessary to note that the quantity tan2δ in the index of the characteristic function determines the bond between displacements of two points of the uneven boundary. As one can see from expression (3.41), in the process of calculation of the average intensity, tan2δ is essential in the exponential index, though we may ignore fluctuations of surface inclination while determining the scattered field (see expression (3.29)). Now let us analyze expression (3.41) and consider two limiting values of the parameter ∆ (∆ > 1). Let ∆ > 1 corresponds to the smallness of the light spot as against the correlation length of the surface unevenness, i.e., the relationship a/ρ0 1 that at kσ >> 1 corresponds to the radius of the light spot comparable to or larger than the correlation length of the boundary unevenness, i.e., the condition a/ρ0 ≥ 1. The average intensity is described by an expression, 2 I 0αmc a2 µρ 0 2 〈| p |〉 ≈ 2C p R [( µ / k ) 2 + cos 2 θ ] 2 kσ
2
× (3.42)
1 ρ 2 exp − 0 tan 2 θ . 4 σ As one can see from formula (3.42), in this case the angular characteristic is determined by the scale of surface unevenness. The intensity decreases e times for the observation angle corresponding to the relationship tanθ = 2σ/ρ0, i.e., the angular beam of the directivity pattern is approximately
60
THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID
equal to the mean-square angle of inclination of the uneven surface θk = 21/2σ. Directivity patterns for the average intensity are given in Fig. 3.4 for µ = k, σ/ρ0 = 0.17, and ∆2 = 0.1, 1, and 10. The limited angular width of the directivity pattern θ 0 = arctan(21/2σ/ρ0) = 14° for the selected ratio σ/ρ0 is indicated at the plot at ∆2 = 10.
Figure 3.4 Polar diagrams of average intensity distribution. Curves 1 – 3 correspond to the next values of the parameter ∆2: 0.1, 1, and 10.
It is easy to determine the dispersion of acoustic field fluctuations using formulae (3.32) and (3.41): 2 I 0αmc 1 µ 2 a 4 cos 2 θ 2 2 × D = 〈| p | 〉− | 〈 p〉 | = 2C p R [( µ / k ) 2 + cos 2 θ ] ∆2 cos 2 θ + 1
ρ 2 a 2 k 2 sin 2 θ ∆2 sin 2 θ exp − − k 2σ 2 cos2 θ . exp − 0 2 σ 4( ∆2 cos2 θ + 1) (3.43) In the case of the dimension of the radiation spot small being compared with the correlation length of boundary unevenness, the dispersion is proportional to the factor [1 − exp(−p/4)], i.e., the dispersion is maximal in the direction of the axis z and increases with the growth of the Rayleigh parameter. If the dimension of the radiation spot is large in comparison with the correlation length, the angular dependence of the dispersion is close to
RADIATION ACOUSTICS
61
the angular dependence of the square of the average pressure. In this case the dispersion is minimal in the direction of the axis z and tends to zero in this direction with the growth of the Rayleigh parameter. According to this consideration, we note the next characteristic features of the sound field arising due to absorption of modulated penetrating radiation in a liquid half-space with large-scale unevenness of the boundary. The average pressure represents the product of the sound pressure in the half-space with even boundary and the characteristic distribution of the height of boundary unevennesses. The average pressure depends essentially on the Rayleigh parameter as in the problems of sound scattering. The influence of boundary unevenness on the average intensity of sound field is given by the exponential dependence on the parameter ∆, i.e., in the case of large unevennesses (kσ >> 1), the intensity depends exponentially on the ratio of the radius of the radiation spot at the boundary to the correlation length of boundary unevenness. If the radius of the radiation spot is small as against the correlation length of unevennesses, we may ignore the effect of boundary unevenness on the average intensity of the field. In this case, the larger the Rayleigh parameter, the larger the peak of the directivity pattern along the axis z of the dispersion. If the radius of the radiation spot is larger than the correlation length of unevenness, the angular width of the directivity pattern is approximately equal to the average mean-square angle of inclination of the uneven surface. In this case the dispersion has a directivity pattern with the minimum along the axis z.
5. THE CASE OF SMALL UNEVENNESS Now let us consider the effect of an uneven liquid boundary on thermoradiation generation of sound when unevennesses are assumed to be mildly sloping, statistically homogeneous, and isotropic and their height is small in comparison with the sound wavelength. We consider the developed generation mode as above. We will obtain the expressions providing an opportunity to calculate the average field and the intensity of sound field fluctuations in the far wave zone. Simple relationships connecting the average sound field with the Rayleigh parameter, mean-square height, and spatial correlation length of unevennesses will be given for some limiting cases. As one will be able to see, the boundary unevenness affects sound field fluctuations in two ways: first, the field of volumetric thermal sources is scattered at random boundary unevennesses, and second, the intensity of these sources fluctuates as the track length of particles (quanta) of penetrating radiation in a liquid changes randomly.
62
THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID
Let a penetrating radiation beam propagating in the positive direction of the axis z be incident upon the free uneven boundary of a liquid half-space z ≥ ξ (x, y). The boundary unevenness is random and ξ ( x, y ) = 0. Here and further, the line above means averaging with respect to the statistical ensemble. Using the reciprocity principle, we write down the solution of the equation of thermoradiation sound generation in the form p(r0 ) =
iωmαAµ Cp
∫ I ( x, y) exp{−µ[ z − ξ ( x, y)]}G(r, r0 )dV ,
(3.44)
V
where G(r, r0) is the function, which is the solution of the inhomogeneous reduced wave equation (the equation of thermoradiation sound generation in a liquid) with the right-hand side in the form of the δ-function, satisfying the condition G (r, r0 ) | z =ξ ( x, y ) = 0 ,
(3.45)
at the uneven surface, where r(x, y, z) is the radius-vector of the current point and r0(x0, y0, z0) is the radius-vector of the observation point. Integration in expression (3.44) is performed over the part of the half-space z ≥ ξ (x, y), where thermal sources exist. The function G(r, r0) is the solution of the problem, i.e., the field of a point source located at the point r0 of the half-space with the uneven boundary, where it is necessary to determine the radiation field. It is known that the exact analytical representation for G(r, r0) has not been obtained yet. Therefore, making certain assumptions on the problem parameters, we use an approximate expression for G(r, r0). We need to determine the sound field in the far wave zone4. In this case the function G(r, r0) can be represented as the solution of the problem on the diffraction of a plane monochromatic wave at an uneven boundary, and the spherical divergence of the field can be taken into account with the help of the common factor exp(ikr0) / (4π r0). We assume that the height of boundary unevennesses is small as against the sound wavelength. We restrict ourselves also for simplicity to 4
As in the previous section, we mean here the far wave zone with respect to both the dimensions of the region of effective heat release (and therefore, effective sound generation) and the uneven boundary. The conditions determining the far wave zone with respect to the uneven boundary have been considered in detail by Lysanov [125] for example.
RADIATION ACOUSTICS
63
consideration of mildly sloping, statistically homogeneous, and isotropic unevennesses of the boundary. The problem of scattering of a plane sound wave at the boundary with small mildly sloping unevennesses has been considered in details by many researchers (see [18, 125] for example). We will basically follow these considerations to determine the solution. Let us expand boundary condition (3.45) into a series with respect to the power of the small parameter kσ, where σ is the mean-square height of unevennesses, and keep in the expansion only the terms of order of smallness not higher than the second. Then the exact boundary condition is changed for the approximate one at z = 0. We represent the field described by the function G(r, r0) in the form of the sum of the average field Gav(r, r0) and the random addition Gran(r, r0), G (r, r0 ) = Gav (r , r0 ) + G ran (r , r0 ) ,
where G ran (r , r0 ) = 0. Let us write down an approximate expression for Gav(r, r0) keeping the terms of the order of smallness not higher than the second order with respect to kσ: Gav (r, r0 ) =
exp(ikr0 ) exp[i (k x x + k y y )] × 4πr0 (3.46)
[exp( −ik z z ) + w exp(ik z z )] , where (kx, ky, kz) are the components of the wave vector k coinciding in its direction with the radius-vector r0 and w is the average coefficient of reflection of a plane sound wave from the uneven boundary, which is represented (as it is possible to demonstrate) with the help of the normalized function of correlation of statistically homogeneous and isotropic unevennesses N(ρ), where ρ is the horizontal distance between two points at the boundary: w = −1 + η cosθ ; 2 ∞ 2 2 ∂ exp(ik ρ + z ) η = 2ikσ 2 ∫ N ( ρ ) J 0 (kρ sin θ ) ρ dρ . ∂z 2 2 2 ρ +x 0 z = 0
(3.47)
Here J0(kρ sinθ) is the Bessel function of the zero order and θ is the angle between the axis z and the radius-vector r0. Thus, we have
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THERMORADIATION EXCITATION: HOMOGENEOUS LIQUID
Gav (r , r0 ) = G0 (r , r0 ) + O(k 2σ 2 ) , where
G0 (r, r0 ) =
exp(ikr0 ) exp[i (k x x + k y y )][exp(−ik z z ) − exp(ik z z )] . 4πr0
(3.48)
Within the framework of the above assumptions, the term Gran(r, r0) has the form Gran (r , r0 ) =
exp(ikr0 ) Ψ ( x, y , z ) , 4πr0
(3.49)
where Ψ(x, y, z) is the random component of the field of scattering of a plane monochromatic sound wave incident at the angle θ on the statistically uneven boundary of the half-space. As is known (see [125]), the function Ψ(x, y, z) can be represented in the form of the double Fourier integral, Ψ ( x, y , z ) = ∫
∞
∫ A(α , β ) exp[i(αx + βy + γz )]dαdβ ,
(3.50)
−∞
where γ = (k2 − α2 −β 2)1/2, and an expression for the random amplitude A(α, β) has the form A(α , β ) =
i 2π 2
k cos θ ∫
∞
∫ ξ ( x, y ) ×
−∞
(3.51) exp{−i[(α − k x ) x + ( β − k y ) y ]}dxdy . Let us make one more assumption. Let µσ > 1, where ρ0 is the correlation coefficient of unevennesses. Let us consider the double integral in expression (3.58). At the essential interval of integration, κ > 1) 3 η = 2(kσ ) 2 cos θ , f (θ ) = 1 − P 2 − ( µσ ) 2 , 8 where P = 2kσ cosθ is the Rayleigh parameter. Thus in this case, the decrease of the average field in comparison with the field in a half-space with an even boundary is determined not only by the Rayleigh parameter as in the case of large-scale unevenness, but also by the quantity µσ. It is necessary to note that the latter does not depend on the observation angle. In the case of slip observation angles such that kρ0 cos2θ 1) with Gaussian profile, it is easy to obtain a general expression for η:
η = (c / ρ )(mAα / 2C p ) 2 [ µk ( µ 2 + k 2 )]2 I 0 .
(3.63)
Thus, the efficiency of radiation-acoustic conversion η depends linearly on the penetrating radiation efficiency I0 = Prad/(πa2). The maximum efficiency corresponds to the condition k = µ and is determined by a formula,
η max = (c / ρ )[mAα / 4C p ] 2 I 0 .
(3.64)
Let us give some estimates of generation of monochromatic sound in water (in many cases the necessary radiation-acoustic and thermophysical constants are well known for water) and in particular, the estimates for the cases of sound generation by CO2- and YAG-lasers. In these cases the thermal expansion coefficient α = 3⋅10−4 K−1 and according to expression (3.64), the maximum conversion efficiency ηmax ≈ 5⋅10−12 W/cm2.
RADIATION ACOUSTICS
73
CO2 laser (λ = 10.06 µm). Let us calculate the maximum value of the ultrasonic amplitude that can be obtained at the axis of a laser beam at a distance R = 1 m from the water surface while using a laser pulse with length 10 µs and modulated with an ultrasonic frequency (intra-pulse modulation). The absorption coefficient µ ≈ 800 cm−1 [99, 119, 179]. At the optimal sound frequency, the far wave field is realized in the case of focusing of laser radiation into a spot a ≤ 2 mm. Assuming a = 1 mm and Popt = 103 W, we obtain according to expression (3.22), p ≈ 500 dyne/cm2 at the sound frequency 6 MHz and in the sound spot ∆l ≈ 15 cm. In this case the intensity I0 = Popt/(πa2) = 3⋅104 W/cm2 and according to expression (3.64), the conversion efficiency η ≈ 5⋅10−8. YAG laser (λ = 1.06 µm). Let us calculate the amplitude of ultrasound emitted in the quasi-CW mode along the water surface at the frequency ω/2π = 100 kHz at distance R = 1 m from the radiation spot of radius a = 2.5 mm. The absorption coefficient µ = 0.18 cm−1. As k ≈ 4 cm−1, then ka ≈ 1, ka2 ≈ 2.5 mm, and k/µ >> 1, i.e., the conditions for sound radiation along the liquid surface are satisfied. In this case the direction of the maximum of the directivity pattern constitutes the angle θ ≈ µ/k ≈ 4.5⋅10−2 rad = 2.6° to the liquid surface. Assuming Popt = 50 W, we obtain p ≈ 1.3 dyne/cm2 or 0.13 Pa according to expression (3.22).
CHAPTER 4
Thermoradiation Excitation of Sound in an Inhomogeneous Medium Inhomogeneity of liquid may influence sound excitation by penetrating radiation. Firstly, parameters of thermoradiation sources of sound may change because of the change of the path of radiation particles in a medium. Secondly, sound waves may be refracted and scattered by inhomogeneities and reflected by boundaries. In this chapter we consider some examples of thermoradiation excitation of sound in an inhomogeneous liquid.
1. SOUND EXCITATION IN A LIQUID HALF-SPACE IN THE PRESENCE OF A LAYER OF ANOTHER LIQUID AT ITS BOUNDARY Let us consider specific features of thermoradiation generation of sound in a liquid half-space in the presence of a layer of another liquid at its boundary, and in the case of absorption of intensity-modulated penetrating radiation in a two-layer medium. The presence of a liquid layer with acoustic (radiation) parameters, which differ from the parameters of the liquid in the half-space, can substantially influence characteristics of sound emission and intensity of the sound field. Both increase and decrease of intensity are possible in this case [145]. Let a beam of penetrating radiation be incident normally upon a free boundary of a liquid layer with density ρ and sound velocity c. Let the layer 75
76 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM
of the thickness H be located on a liquid half-space with density ρ1 and sound velocity c1. It is necessary to determine the sound field in the halfspace produced by absorption of intensity-modulated penetrating radiation in such a two-layer medium. As usual we select a coordinate system in such a way that the plane (x,y) coincides with the free surface of the layer. The axis z is directed within the liquid, and the coordinate origin is positioned in the center of the place of incidence of penetrating radiation (radiation spot) at the free surface of the liquid. We assume that the intensity of penetrating radiation is lower than the value necessary for changes of aggregate state in the region of absorption. Let us consider the stable regime of sound generation. The field of sound pressure p in the layer is described by the solution of the equation, ∆p + k 2 p =
iωmα µE ( x, y , z ) , 0 < z < H , Cp
(4.1)
where ω is the frequency of generated sound, k = ω /c is the wave number, m is the modulation index of intensity of penetrating radiation, α, Cp, and µ are the coefficient of cubical thermal expansion, specific heat capacity, and absorption coefficient of penetrating radiation in the layer. The function E(x,y,z) describes distribution of intensity of penetrating radiation in the liquid layer. The sound pressure p1 in the half-space is described by the solution of the equation ∆p1 + k12 p1 =
iωmα1 µ1E1 ( x, y, z ) , z > H , Cp
(4.2)
1
where k1 = ω /c1 and α1, Cp1, and µ1 are the coefficient of cubical thermal expansion, specific heat capacity, and absorption coefficient of penetrating radiation in a liquid occupying the half-space z > H, respectively. The function E1(x, y, z) describes distribution of intensity of penetrating radiation in the half-space. Solutions of equations (4.1) and (4.2) must satisfy the boundary conditions p = 0, p = p1, z = 0,
1 ∂ρ 1 ∂ρ1 = , z = H, ρ ∂z ρ1 ∂z
RADIATION ACOUSTICS
77
and the condition of termination at infinity. The desired solution p1 of equation (4.2) can be written down on the basis of the reciprocity principle. The mathematical formulation of the reciprocity principle in acoustics was obtained by the author [131] (see Chapter 3, Section 2) under very general assumptions for the case of an inhomogeneous medium with constant density. An opportunity to generalize relationships obtained for the case of media with density depending on coordinates was indicated there also. In the case of a twolayer medium, such generalization is attained in a rather simple way described by the author [131]. Therefore, we will not dwell on this technique. Let us write down an expression for the desired sound pressure p1(r0) in the form
ρ iωmα p1 (r0 ) = 1 µ ∫ E ( x, y, z ) p * (r, r0 )dV − ρ Cp Vl
(4.3) iωmα1 µ Cp
∗
∫ E1( x, y, z ) p1 (r, r0 )dV ,
Vhs
where r0(x0, y0, z0) is the radius-vector of the observation point and r(x, y, z) is the instant radius-vector. Integration in expression (4.3) is performed over the layer region Vl and the region of half-space Vhs, where thermal sources of sound exist. The functions p* and p1* describe the field of a point source of unit amplitude located in the observation point r0. Thus, the problem is reduced to determination of these auxiliary functions, and the functions E(x, y, z) and E1(x, y, z) describing the distribution of the intensity of penetrating radiation in the layer and in the half-space. Firstly let us obtain an expression determining the intensity distribution of penetrating radiation in the layer E(x, y, z). In order to do this, it is necessary to sum all reflections of the initial beam of penetrating radiation from the layer boundaries. Thus, a direct beam produces at some level z in the layer the distribution of radiation intensity, which is determined by the expression AI(x, y) exp(−µz), where A is the coefficient of penetrating radiation transmission through the free boundary of the layer, and the function I(x, y) describes the transversal distribution of radiation intensity in a beam. The radiation intensity in the beam at the level z after partial reflection at the boundary “layer – half-space” is described by the expression A(1 − A1 ) I ( x, y ) exp[− µ (2 H − z )] ,
78 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM
where A1 is the transmission coefficient of radiation through the boundary “layer – half-space”. Then, after a partial reflection from the free boundary of the layer, the radiation intensity at the level z decreases down to the value A(1 − A)(1 − A1 ) I ( x, y ) exp[− µ (2 H + z )] , and so on. It is easy to sum the series obtained using the formula for the sum of an infinitely decreasing geometric progression. In the result we can write down an expression for E(x, y, z): E ( x, y , z ) = A
exp(− µz ) + (1 − A1 ) exp[− µ (2 H − z )] I ( x, y ) , 1 − (1 − A)(1 − A1 ) exp(2 µH )
(4.4)
at 0 ≤ z ≤ H. The intensity of penetrating radiation in the half-space is calculated in an analogous way. A corresponding expression for E1(x, y, z) can be presented in the form
E1 ( x, y, z ) =
AA1 exp[− µH − µ1 ( z − H )] I ( x, y ) , z ≥ H . 1 − (1 − A)(1 − A1 ) exp(−2µH )
(4.5)
It is necessary to determine the excited sound field p1(r0) at the point r0 in the liquid half-space at distances greater than the dimensions of the region occupied by thermal sources of sound. In this case we assume that the observation point r0 is located not too close to the boundary “layer – half-space”1. In this case the auxiliary solutions within this area can be represented approximately in the form
p * (r , r0 ) =
1
exp(ik1r0 ) exp(−ik1 x sin θ )W [exp(−iγz ) − exp(iγz )] , 4πr0
We have to note that under certain conditions, a normal mode relatively weakly emitting energy into the half-space can exist in the layer, and therefore it relatively weakly attenuates with distance from the source. The solution given below is valid also in the case when the distance from the source to the observation point is large in comparison with the dimensions of the layer region, where the existence of a weakly attenuating normal mode is noticeable.
RADIATION ACOUSTICS
p * (r , r0 ) =
exp(ik1r0 ) exp(ik1 x sin θ ){exp(−ik1 z cos θ ) − 4πr0
79
(4.6)
V exp[ik1 ( z − 2 H ) cos θ ]} . Here W =
2 β exp[i (γ − k1 cosθ ) H ] 1 − β + (1 + β ) exp(2iγH ) , V = , 1 + β + (1 − β ) exp(2iγH ) 1 + β + (1 − β ) exp(2iγH ) (4.7)
γ = k 2 − k12 sin 2 θ , β =
ρk1 cosθ , ρ1γ
θ is the angle between the radius-vector r0 and the axis z, and the plane x, y is assumed to include the observation point. Substituting auxiliary solutions in the form of expressions (4.6) and E(x, y, z) and E1(x, y, z) in the form of expressions (4.4) and (4.5) into expression (4.3), we obtain a relationship determining the sound pressure p1(r0):
p1 (r0 ) =
ρ α WµγQ exp(ik1r0 ) Aωm + F (θ ) 1 1 − (1 − A)(1 − A1 ) exp(−2µH ) 2πr0 ρ C p µ 2 + γ 2 (4.8)
µ1k1 cos θ (1 + V ) + iµ12 (1 − V ) α 2 A1 exp[− µH − ik1H cos θ ] , C p1 2 µ12 + k12 cos2 θ where
F (θ ) = ∫
∞
∫ I ( x, y) exp(ik1 x sin θ )dxdy ,
−∞
Q = 1 − 2 exp(− µH ) cos(γH ) −
80 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM
µ A1 exp( − µH ) sin(γH ) − cos(γH ) − (1 − A1 ) exp( −2 µH ) . γ Expression (4.8) determines the sound field in a liquid half-space in the far wave zone. The field is produced due to absorption of intensity-modulated penetrating radiation in the case when a layer of another liquid is present at the half-space boundary. Using this expression, we can calculate directivity patterns of thermoradiation sources of sound in various situations. As we have noted before, a transverse distribution of intensity in a beam of penetrating radiation is close to the Gaussian one. Therefore, we assume I ( x, y ) = I 0 exp[−( x 2 + y 2 ) / a 2 ] . Integrating, we obtain the next expression for F(θ ): F (θ ) = I 0πa 2 exp[−(k12 a 2 / 4) sin 2 θ ] .
(4.9)
Figure 4.1 shows examples of directivity patterns calculated according to formulae (4.8) and (4.9) for a laser source (photon beam) and the Gaussian distribution of intensity in a laser beam. Directivity patterns in Fig. 4.1a correspond to the situation in which a layer of benzene is present at the surface of water. The wavelength of optical radiation is 0.3 µm. In this case µ = 2.3 cm−1 and µ1 = 0.18 cm−1 (for example, see [179]). The coefficients of light transmission are A = 0.96 and A1 = 0.99. The radius of light spot is taken to be equal to a = 0.4 cm. The ratios are ρ/ρ1 = 0.88, α1/Cp = 63.3⋅10−12 g/erg, and α1/Cp1 = 4.7⋅10−12 g/erg. The directivity patterns are calculated for the three values of thickness of the benzene layer: H = 0.33 cm (kH = π/2), H = 0.66 cm (kH = π), and H = 1.99 cm (kH = 3π). The observation angle is counted off from the vertical axis, where the modulus of the amplitude of sound pressure p1(r0) is normalized to (2ωmI0a2/2r0)⋅10−12 g/(cm⋅s2). We should note that under such conditions the directivity pattern of a laser thermoradiation source in water without a benzene layer is strongly extended along the surface, and the amplitude of sound pressure in the vertical direction (θ = 0) is approximately two orders of magnitude smaller than that in the considered case with a layer. Thus, the presence of a layer of another liquid on the surface of a liquid half-space can lead to a significant change of intensity of sound field. If a liquid in the layer is characterized by the value of α /Cp large in comparison with a liquid in the half-space, the intensity of sound field increases
RADIATION ACOUSTICS
81
substantially. The difference in the ratios µ/k for liquids in the layer and in the half-space leads also to significant differences in directivity patterns in the cases with a layer and without it. As we have noted already, in the absence of a layer, the directivity pattern is strongly extended along the free surface of water. And when a benzene layer is present, it displays a peak in the vertical direction. Finally, if the values of kH are large enough, the resonance properties of a layer begin to manifest themselves which also leads to changes (irregularity) in directivity patterns.
Figure 4.1 Directivity pattern of a thermooptical source of sound in a two-layer medium. (a) Benzene – water, thickness of benzene layer H = 0.33, 0.66, and 1.99 cm (curves 1 – 3, respectively); (b) water – benzene; and (c) warm water – cold water correspond to the thickness H = 0, 1.86, and 2.23 cm (1 – 3).
Directivity patterns given in Fig. 4.1b describe a reversed situation: a water layer is located on a benzene half-space. Calculations are conducted
82 THERMORADIATION EXCITATION: INHOMOGENEOUS MEDIUM
for three values of layer thickness, i.e., H = 0 (a half-space without a layer), H = 1.86 cm (kH = 2.5π), and H = 2.23 cm (kH = 3π). Such arrangement of liquids may be of theoretical interest only. However, it is important to note the following fact. Only about 30% of the energy of laser radiation is absorbed in a water layer in this case. Moreover, the ratio α/Cp for benzene is more than one order of magnitude larger than this ratio for water. Therefore, acoustic radiation is produced almost exceptionally by thermal (thermooptical) sources of sound located in the half-space, i.e., benzene. Nevertheless, the shapes of directivity patterns in the cases with a layer and without it are essentially different. This fact is connected with the change of conditions at the boundary of the half-space: a layer is present with wave resonance properties instead of a free boundary. Directivity patterns given in Fig. 4.1c correspond to a situation in which a layer of warm water (20°C) is located on top of cold water (0°C). The wavelength of laser radiation is selected to be equal to 1 µm. In this case µ = µ1 = 0.18 cm−1. The frequency of generated sound is ω /2π = 105 Hz. The next values of wave numbers are k = 4.23 cm−1 and k1 = 4.38 cm−1. Furthermore, A = 0.98, A1 = 1, ρ/ρ1 ≈ 1, α/Cp = 4.76⋅10−12 g/erg, and α1/Cp = 1.19⋅10−12 g/erg. The radius of the light spot is taken to be equal to 0.4 cm. The patterns are calculated for three values of layer thickness: H = 0 (a homogeneous half-space), H = 1.86 cm (kH = 2.5π), and H = 2.23 cm (kH = 3π). In the case considered, the liquids in the layer and the half-space differ noticeably only in the ratio α /Cp. Nevertheless, the presence of the layer leads not only to an increase of the intensity of generated sound, but also changes significantly the shapes of directivity patterns. It is difficult to conduct a more detailed analysis of the dependence of the parameters of the sound field generated in a two-layer medium by a source of penetrating radiation on the problem parameters in a general form, as expression (4.8) is rather complex. Therefore, we restrict ourselves to consideration of a particular case which may be of practical interest. Let exp(−µH) 1, the maximum value of q is attained at kH = (n + 1/2)π and is equal to 2 2 ρ c µ + k1 . q= 1 1 ρc µ 2 + k 2
It is necessary to note that the value of q may be larger than one in this case. Everything discussed is illustrated by the directivity patterns calculated according to approximate formula (4.10) and given in Fig. 4.2. Calculations were conducted for the Gaussian distribution of the intensity of penetrating radiation in a beam (in a laser beam) as above. The observation angle was counted off the vertical axis where the level of sound pressure p1(r0) normalized to the value of
ωmαA 2 I 0 µk a 2 Cp 2r0 µ + k 2 is displayed. The next relationships between the problem parameters are taken: ρ/ρ1 = 0.79, µ/k1 = 1, and ka = 2. Directivity patterns in Fig. 4.2a correspond to the value k/k1 = 1 and in Fig. 4.2b k/k1 = 1.27. These directivity patterns were calculated for the values kH = 2.5π and kH = 3π. The directivity pattern of sound radiation generated by a radiation-acoustic source in a homogeneous half-space without a layer is given for comparison. We have to note that the values k/k1 = 1.27 and ρ/ρ1 = 0.79 correspond roughly to the case when a layer of alcohol is present on the water surface. One more particular case is interesting. Let ρ = ρ1 and k2 = k12 + ib2, while b/k1 > 1. Then, as is demonstrated by Lyamshev and Sedov [144], f(θ) = 1 – 3P2/8 – (µσ)2, where P = 2kσ cosθ is the Rayleigh parameter and σ is the mean-square-root height of unevenness. In this case the relationship v = − 1 + P2/2 is valid for the average reflection coefficient. Substituting these values of f(θ) and v into expression (4.14) and keeping the terms of the order of smallness not higher than the second order with respect to kσ and µσ, we obtain an expression, 3 p2 (1 − β ) exp(2iγH ) pav (r0 ) = p1 (r0 )1 − p 2 − ( µσ ) 2 + , (4.15) 2 1 + β + (1 − β ) exp(2iγH ) 8 (expression (4.10) is used here for p1(r0)). Expression (4.15) is obtained under the assumption p2 (1 − β ) exp(2iγH ) k = 42 cm−1 is given in Fig. 4.3b. One can see that the “gaps” in angular patterns not only attain the value of sound pressure in the case of a free boundary, but are also much deeper. Figures 4.3c and d illustrate the cases µ 2/(a sinθ ). Therefore, we can ignore the quantity k2 as against µ2/cosθ in the denominator of the integrand. Integrating (see [21], p. 73 (19)) we obtain an expression for sound pressure at s >> 1 in the form R/c −t 2 p ( x, y , z , t ) = ( R − ct ) exp − 2 2 π C p Rs sin θ τ a2 I 0αc
− (5.7)
R / c − t + τ 2 . ( R − ct + cτ ) exp − 2 τa Sound pulses determined by expression (5.7) are close to the ones given by curves 3 in Fig. 5.2a and b. The pulse shape is determined by the characteristic times τ and τa in this case. In the case of small τ satisfying the condition cτ τ = > 1 the shape and length of sound pulses are determined completely by the characteristic time τa. The pressure amplitude is inversely proportional to the parameter s.
RADIATION ACOUSTICS
111
If s >> 1, which corresponds to the region of thermal sources in the shape of a narrow cylinder or to observation under small angles θ, then the decrease of the integration element in expression (5.5) with growth of frequency is determined by the denominator, and for k >> µ/cosθ the integration element becomes close to zero. Therefore, the exponent in the integration element can be changed to one. Integration gives an expression for sound pressure in the case s τa, τµ, the length of a sound pulse is determined by the length of a radiation pulse independently of the change of the observation angle θ and the ratio of the characteristic times s. In this
112
EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
case the pressure amplitude decreases as the observation angle (i.e., the parameter s) grows.
Figure 5.3 Levels of sound pressure (relative) in water at s > 1 the shape and length are determined by the value of τa). In particular this is also true for observation along the axis of the radiation beam. We can determine the time-average intensity of pressure field for pulses repeated with period T. The pulse-average field intensity averaged with respect to period T is evidently the time-average intensity and it is represented by the next expression:
p
2
∞
2 2 2 ≡ p = ∫ F [ p] dω , T T 0
(5.13)
where 〈|F[p]|2〉 is the spectral power average with respect to surface realizations. Let us consider as an example the normal law of distribution of heights of unevennesses large in comparison with the sound wavelength, i.e., the case kσ >> 1. Using the results obtained in Chapter 3, Section 4 and taking into account the remarks on the inverse Fourier transform, it is possible to write down the time-average intensity:
p
2
1 I 0 µα = T C p R
2 a 4c ∞ 1 − cos(kcτ ) × cos 2 θ ∫ ( µ 2 / cos 2 θ + k 2 ) 2 0
(5.14) 2 ∆2 sin 2θ k 2 dk ρ0 exp − , 2 2 2 2 2 4(1 + ∆ cos θ ) 1 + ∆ cos θ 2
where ∆ = 21/2kaσ/ρ0 is the parameter of scattering and ρ0 is the correlation coefficient of boundary unevenness. As has been demonstrated in Chapter 3, Section 4, at ∆ > 1 (the radius of radiation spot at the liquid surface is comparable to the correlation coefficient of unevenness of liquid boundary or larger), the effect of unevenness on the average intensity of sound field is essential and we obtain the next formula for the time-average intensity:
p
2
=
π T
2 1 tan θ I 0α a 2c 1 exp− C p R 4µ cos θ tan 2 β 2 tan δ
2
× (5.15)
τ 1 − 1 + τ µ
exp − τ τµ
,
where tanβ = 21/2σ /ρ0 is the mean-square angle of surface slope. The average intensity of field (5.15) in the case of large radiuses of penetrating radiation beam (a/ρ0 ≥ 1) has an angular distribution. The angular width of the lobe of a polar directivity pattern is equal to 21/2β and determined by the mean-square angle of surface slope β. The average intensity depends essentially on the ratio τ/τµ: if the length of radiation pulse is small τ 0. Let the beam intensity at the liquid boundary be described by an expression, I ( x, y , t ) = I ( x, y ) f (t ),
(5.16)
where f(t) is the function of time, which determines the shape of a radiation pulse in such a way that max f(t) = 1, and I(x, y) is the surface distribution of radiation intensity at the boundary. The solution of the reduced wave equation for the spectrum of sound pressure generated by penetrating radiation may be written down in the form of an integral,
pω ( x, y, z ) =
1 4π
iAα ωµI ( x , y ) × p0 ( x, y, z; x1 , y1 , z1 ) − 1 1 Cp V1
∫
(5.17) exp( − µz1 ) F (ω )dV1 , where p0 is the solution of the diffraction problem for a point source, V1 is the volume occupied by thermal sources of sound, which are produced by the action of penetrating radiation, and ∞
F (ω ) =
∫ f (t ) exp(iωt )dt
−∞
118
EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
is the spectrum of radiation pulse. A solution of the stationary problem for a liquid half-space with a free boundary can be obtained readily in the far wave field for a specific distribution of intensity I(x1, y1). Let us consider the Gaussian distribution of intensity as usual, I ( x1 , y1 ) = I 0 exp[−( x12 + y12 ) / a 2 ] , where a is the effective beam radius. In this case the spectrum of sound pressure has the form
pω = −
2 2 2 AI 0αa 2 e ikR ω τ µ e −ω τ a / 4 F (ω ) , 2C p R 1 + ω 2τ 2 µ
(5.18)
where R is the distance from the center of the region of thermal sources to the observation point, τµ = cosθ/(µc) and τa = a sinθ/c are the characteristic delay times of sound from elementary sources in the vertical and horizontal sections of the region of thermal sources, respectively, θ is the angle between the axis z and the direction to the observation point, and c is the sound velocity. An expression for sound pressure is obtained by the inverse Fourier transform of spectrum (5.18),
p=−
∞
κ 2s2 exp − 4 4πC p Rτ µ2 − ∞ AI 0αa 2
∫
κ2 κ exp(iκγ ) F τµ 1+ κ 2
dκ ,
(5.19)
where the substitution κ = ωτµ, γ = (R/c – t)/τµ, and s = τa/τµ is performed. Expression (5.19) is the initial one for further analysis. Let us determine now the characteristic features of acoustic field without setting a specific shape of radiation pulse or its spectrum. Let τ be the length of radiation pulse, which is determined in a certain way, e.g., according to the drop of pulse envelope, the portion of energy contained in a pulse, etc. Let us represent the integral in expression (5.19) in the form of the sum of two integrals,
RADIATION ACOUSTICS
p=−
∞ κ 2s2 AI 0αa 2 − exp 2 ∫ 4 4πC p Rτ µ − ∞
exp(iκγ ) F κ τµ
119
dκ − (5.20)
∞
κ 2s2 κ ∫ exp − 4 exp(iκγ ) F τ µ −∞
dκ . 1+κ 2
The spectral width of functions in the integration element in expression (5.20) can be determined in the following way: the upper frequency limit for the spectrum of radiation pulse F(ω) is obtained on the basis of a known relationship [182] ω = c1/τ, the upper limiting frequency for the exponential function exp(−κ2s2/4) is ω = c2/τ, and the frequency limit for the rational function 1/(1 + κ2) is ω = c3/τµ, where c1, c2, and c3 are the constants depending on the way of determination of the spectrum width and the width of radiation pulse. Thus, depending on the relation of the characteristic parameters of the problem (τ, τa, τµ), some function under the integral determines the character of the drop of spectral density. Let us consider the limiting relationships τ >> τa and τ > τa, i.e., we consider the region of thermal sources in the form of a narrow cylinder or radiation along the directions close to the axis z. Under this condition we may assume that the exponent exp(−κ2s2/4) in expression (5.20) is equal to one within the frequency range, where the spectrum of a pulse of penetrating radiation is essential, and write down an expression
p=−
∞ AI 0αa 2 κ R τ f t F − − ∫ 2 µ τ c 4πC p Rτ µ −∞ µ
exp(iκγ ) dκ . 1 + κ 2
(5.21)
The second integral can be estimated for the cases τ >> τµ and τ > τµ, i.e., the drop rate of spectral density in integral (3.6) determines the spectrum of a pulse of penetrating radiation F(ω), the rational function can be expanded in a series and integrated:
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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
∞
κ ∫ F τ µ −∞
∞ ∞ R exp(iκγ ) (−1) n κ 2n dκ = τ µ2n +1 f 2n t − . ∑ ∑ c n=0 n=0
Let us assume as has been agreed earlier that the series converges with respect to the power of the parameter τµ /τ τ, s > 1). In the second case when θ 0 = θ 1 = θ 2, expression (5.24) is true for the angles θ < θ 0 (τ > τa, τ < τµ) and expression (5.27) is valid for θ > θ 0 (τ < τa, s > 1). In the third case at θ < θ 1 (τ > τa, τ < τµ), formula (5.24) is true, expression (5.22) is valid at θ 1 < θ < θ 2 (τ > τa, τ > τµ), and at θ > θ 2 (τ < τa, s < 1) expression (5.27) is true.
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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
Figure 5.6 Ranges of application of expressions (5.22), (5.24), (5.27), and (5.28). (a) τ < τ0; (b) τ = τ0; and (c) τ > τ0. Digits in brackets are the numbers of expressions.
We have to note in conclusion that the conducted theoretical consideration can be extended to the case of a liquid with rough surface using the results obtained in Section 2 of this chapter.
4. NEAR WAVE FIELD OF THERMORADIATION PULSED SOURCE OF SOUND In the previous section we have discussed the characteristics of acoustic radiation far from the region of sound generation. Meanwhile, the situation when the track length of penetrating radiation in a liquid is large enough and detection of an acoustic signal occurs in the near wave field of a thermoacoustic source arising due to absorption of penetrating radiation in a liquid, is quite common. Such a case is considered in this section (see [190]). Let a beam of penetrating radiation be incident perpendicularly on the free surface of a liquid, and the radiation track length l = µ−1, where µ is the coefficient of radiation absorption in the liquid, be large compared with the radius of the beam cross-section a (l >> a). We assume that the intensity distribution of penetrating radiation is even (and not Gaussian as usual) over the cross-section. A cylindrical region emitting a sound wave due to thermal expansion is formed in the liquid under the action of radiation pulse (Fig. 5.7). Taking equality of sound pressure to be zero as the boundary condition at the free surface of a liquid, we can write down the solution of the wave equation of thermoradiation generation of sound in the form p=
1 α 4π C p
1 ∂Q(t − r1 / c, r ) dV − ∂t
∫ r1
V
RADIATION ACOUSTICS
−
1 α 4π C p
1 ∂Q(t − r2 / c, r ) dV . ∂t
∫ r2
V
125
(5.29)
We assume that the length of radiation pulse τ is so large that cτ >> a, and we can ignore the difference between the arrival times of signals from different points of the beam cross-section.
Figure 5.7 To calculation of the near wave field of a pulsed thermoradiation source. (1) A beam of penetrating radiation; (2) the surface of a liquid; and (3) a thermoradiation source.
Let us consider the characteristics of a sound field in the near wave field of the radiating region, i.e., at distances x > a, µcτ = 22.5. The acoustic signal at the point A starts at the moment t1 = xA /c when a compression wave from the closest region of sources arrives at this point. Then, at the moment t2 ≈ (xA2 + zA2)1/2 a rarefaction wave arrives. This wave is produced by addition of the pulse radiated by sources located in the vicinity of the point 0 and the wave reflected from the free surface. The rarefaction wave is added to the compression wave. Therefore, the pulse length depends on the difference of arrival times of the compression and rarefaction waves. This difference is determined by the relationship
τ ' ≈ t2 − t1 .
(5.32)
Oscillograms of an acoustic signal excited by a laser pulse in water and detected at the point with the coordinates xA = 10 cm, zA = 12 cm under the conditions corresponding to the calculation are given in Fig. 5.10a and b.
Figure 5.10 Oscillograms of an acoustic signal at the observation point with the coordinates zA = 12 cm, xA = 10 cm. (a) A signal in a wide frequency band; (b) a signal in a low frequency band.
The upper picture demonstrates a signal detected by a wide-band receiver. One can see that the acoustic pulse consists of a leading and a high-frequency component, which are specified by the spiking structure of
128
EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
the laser pulse. The lower picture gives the low-frequency part of the signal separated by a filter. As one can see, the leading part of the pulse is similar to the theoretical dependence describing the shape of an acoustic pulse in the near wave field of a thermoacoustic array. If the distance from sound sources in the direction perpendicular to the axis of the laser beam grows, the difference between the arrival times of compression and rarefaction waves decreases and they eliminate each other. As the result of this process under the condition τ1 < τ′ > zA, where τ = 0.
Figure 5.11 Oscillogram of an acoustic signal close to the surface at xA > zA.
A different pattern is observed in the case when the difference of the arrival times of compression and rarefaction waves τ′ exceeds the characteristic length of a radiation pulse τ. Such situation may arise in the case of short radiation pulses or in the case of moving away from the observation point along the axis z. Figure 5.9b gives as an example the shape of an acoustic pulse at the same observation point as in Fig. 5.9a in the case of a shorter laser pulse when τ′ = 7.5τ. If the distance from the generation region increases, the amplitude of a leading pulse decreases proportionally to exp(−µz) and the amplitude of a closing pulse decreases proportionally to 1/(x2 + z2)1/2.
RADIATION ACOUSTICS
129
If a medium is spatially inhomogeneous and its properties change noticeably at distances smaller than the value of cτ, the structure of an acoustic pulse at the observation point becomes even more complex. For example, let the properties of a medium at the depth z = H change in such a way that the parameter µαS/Cp increases. Then, an additional signal arises at the observation point at the time moment t = (xA2 + H2)1/2/c. The leading part of this signal is a compression wave with amplitude proportional to the degree of medium inhomogeneity. We should note in conclusion that the characteristic properties of the sound pulses excited by pulsed penetrating radiation in the near wave field of a thermoacoustic array can be revealed in the process of consideration of the next simplest one-dimensional model (see [141] and also [41, 210, 215]). Let the energy E be released in the spherical region of the radius R0 during the time τ. This leads to the increase of the volume V = (4π /3)R03 by the value ∆V =
Eα , ρC p
(5.33)
where as usual α is the cubical coefficient of thermal expansion of a medium, ρ is the medium density, and Cp is the specific heat capacity. If R0 >> cτ (c is the sound velocity), the region does not have time to expand during the time of energy release and pressure in the region increases by the value
p = ρc
2 ∆V
V
=
c 2 Eα . C pV
(5.34)
Thus, a spherical region of increased pressure arises in a medium at the initial moment. This leads to radiation of a spherical sound wave. The profile of such wave is given in Fig. 5.12a and the peak value of pressure is equal to a half of the initial excessive pressure multiplied by the ratio R0/r, which takes into account spherical divergence of the wave:
p=
PR0 3 Eαc 2 . = 2r 16π C R 2 r p 0
(5.35)
In this case the pressure amplitude is determined by the released energy and does not depend on its release rate, while the pulse length is determined
130
EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
mainly by the travel time of sound over the disturbed region (the time of volume discharge) R0/c. If R0 > cτ; (b) R0 l in the sense that the first of them corresponds, as it were, to the problem with the initial conditions, and the second one corresponds to the problem with the boundary conditions. In particular, in the case of short radiation pulses (cτ < l), the excessive pressure P arises as a result of energy evolution in a liquid. This pressure is proportional to the increase of the volume ∆V of the region of heat discharge, which is caused by the effect of thermal expansion P = ρc 2V −1dV .
(5.37)
Here ρ is the liquid density, c is the sound velocity in the liquid, V = lS is the volume of the medium where energy is released, S is the sectional area of a radiation beam, ∆V = αE / ρC p ,
(5.38)
where α is the coefficient of thermal expansion, E = IτS is the energy of radiation pulse, and Cp is the specific heat capacity. If the medium-liquid interface is transparent and acoustically rigid, the energy absorbed in the medium region V is radiated in the form of a compression pulse with amplitude proportional to the energy of radiation pulse [110]:
p=
P ρc 2 ∆V c 2 Eα , = = 2 2 V 2lSC p
(5.39)
and its length is determined by the travel time of sound over the region of energy release T = l/c.
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EXCITATION OF SOUND IN A LIQUID BY RADIATION PULSES
In the case of long pulses (cτ > l), continuous generation of sound from the liquid volume V occurs in the process of absorption of penetrating radiation, and the pressure in a sound wave is determined by the expansion rate of the region V, which is proportional to the intensity of penetrating radiation [110], p = ρcV = ρc
∆V αIc . = Sτ Cp
(5.40)
The coefficient of thermal expansion in liquids is small and therefore, the intensity of generated sound can change strongly in the presence of a large amount of gas bubbles affecting noticeably the degree of expansion of a heated volume. The effect of bubbles is determined mainly by their size and concentration in the volume absorbing penetrating radiation. Small bubbles, which are always present in liquids, influence under real conditions the process of thermoradiation generation of sound. This effect plays an essential role even at very small concentrations of bubbles and small (in comparison with the heat of vaporization) density of released energy, when penetrating radiation scattering (caused by bubbles) from the region of energy release is negligible, and the increase of pressure of gas and vapor in a bubble resulting from its heating is small. It is caused by a strong change of compressibility of a liquid containing gas bubbles. Large bubbles have a low resonance frequency f0 and during the time of radiation energy release τ, they behave as incompressible ones. This makes a medium apparently more rigid in the region of heat evolution and increases the amplitude of acoustic pulse. Small bubbles (f0 >> 1/τ) have enough time to change their volume under the effect of increasing pressure and by virtue of large compressibility of gases as against compressibility of liquids, the pressure in an emitted compression wave decreases. The degree of bubble compression may be determined from the energybalance equation: the work of external pressure on a bubble A is equal to the increase of internal energy of gas in a bubble W. The quantity A is expressed by the formula, A = ( P0 + P)(V0 − V1 ) ,
(5.41)
where P is the excessive pressure in a compression wave, which is determined by formulae (5.37) and (5.38), V0 and V1 are the initial and final values of the bubble volume, and P0 is the hydrostatic pressure. The value of W for an adiabatic process is equal to
RADIATION ACOUSTICS
W =
QV0 V0 γ − 1 V1
γ −1
− 1 ,
133
(5.42)
where Q is the initial gas pressure in a bubble and γ is the adiabatic exponent. Equating expressions (5.41) and (5.42), we obtain an equation for determination of the degree of bubble compression z = V0/V1 in the form ( P0 + P)(γ − 1) z γ − z . = Q z −1
(5.43)
If the change of the radius of each bubble is small, i.e., z = 1 + ε, where ε R0), the process of bubble compression is isothermal and we have to take γ = 1 in the expression for ε. Now it is possible to determine the amplitude of a sound wave emitted in a liquid with bubbles. Let M be the volume of gas phase in the generation zone. Then, the change of this volume under the effect of excessive pressure at moderate values of intensity of penetrating radiation is Mε. Thus, the absorbing volume increases by the value ∆V – Mε. Let cτ > l, then using formulae (5.38), (5.40), and (5.44), we obtain pc = ρcc
∆V − Mε . Sτ
It follows from here that pc = αcc I
1 C p (1 + ρcc ⋅ 2nl / τγQ)
.
(5.46)
The dependence of the amplitude of acoustic pressure in a wave emitted from the generation zone on the volume concentration of gas bubbles n for cτ > l is plotted in Fig. 5.13 (it is assumed that f0 >> 1/τ in both cases). The values of pc are normalized to the amplitude of acoustic pressure p in a single-phase liquid to make the plot more illustrative. The next values of constants are taken: 1/l = 0.18 cm−1 and c = 1.5⋅105 cm/s.
Figure 5.13 Dependence of sound pressure amplitude on the volume concentration of gas bubbles. (1) τ = 1 µs, short pulses; (2) τ = 1 µs, long pulses.
The plot illustrates the effect of bubbles on the process of generation of a compression pulse. One can see that the effect becomes noticeable if the concentration of small enough bubbles is not too small.
RADIATION ACOUSTICS
135
Now let us consider the case when the action of penetrating radiation leads to generation of a wave, which has a rarefaction pulse together with a compression pulse. For example, let the energy E be released as the result of absorption of a radiation pulse in a spherical region of radius a during the time τ τχ is satisfied also, expression (7.13) is reduced,
σ RR =
(3 − 4 / n 2 )αa 2 I 0 s R M (θ ) sgn t − 2 c l 8cε Rτ µ
K (θ ) × (7.13a)
t − R / cl exp − τµ
.
And if the condition τχ >> τµ is satisfied, expression (7.13) is reduced as follows:
RADIATION ACOUSTICS
σ RR =
177
τµ (3 − 4 / n 2 )αa 2 I 0 s K (θ ) × M (θ ) + τχ 8cε Rτ χ2 (7.13b)
t − R / cl R sgn t − − 1 exp − τµ cl
.
(c) If τχ τa), the shape of sound pulses does not depend on the parameter τa and is determined by the envelope of radiation pulse, the parameter τµ, and also the relation between the quantities K(θ) and M(θ) (see expressions (7.18) and (7.19)). Everything said above with respect to sound pulses produced by longitudinal waves is applicable to sound pulses produced by transverse waves with one addition. In the case of the observation angles θ > arcsin (1/n) in situations analogous to those described by expressions (7.14) and (7.19), oscillating components appear in sound pulses, the rolloff of the envelope of these oscillations not being taken into account by expressions (7.14) and (7.19). This roll-off must be determined from complete expressions (7.9). The conditions of pairwise equality of the characteristic times τ, τa, τµ, and τχ determine pairwise the characteristic angles of the problem
θ 0 = arctan( µa ) , τ a = τ µ ; θ1 = arccos( µclτ ) , τ = τ µ ; θ 2 = arcsin( clτ / a ) , τ = τ a ; θ 3 = arcsin( χ / acl ) , τ χ = τ a .
RADIATION ACOUSTICS
183
The next six cases are possible: (a) τχ > τ > τ0; (b) τχ > τ = τ0; (c) τχ < τ < τ0; (d) τχ < τ = τ0; (e) τ0 < τχ < τ; and (f) τ < τχ < τ0; where τ0 = (a/cl)(1 + µ2a2)−1/2. In this case the ranges of application of the expressions obtained above for each of the six considered cases and different observation angles θ are given in Fig. 7.1.
Figure 7.1 Ranges of applicability of expressions (7.11) — (7.20) for different observation angles. Digits in parentheses correspond to the expression numbers.
In the case of irradiation of a metal surface by laser pulses, i.e., when the depth of radiation penetration into a substance is small and determined by the skin effect, sound pulses produced by such radiation in both the cases, when the aggregate state of the metal changes and when it does not, have been given by Hutchins, Dewhurst, and Palmer [222]. In conclusion we should note that if we take M(θ) = 0, K(θ) = 2, and τχ = 0 in expressions obtained for sound pulses produced by longitudinal
184
PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS
waves and take into account necessary changes in notations, these expressions coincide completely with the analogous expressions obtained above (see Chapter 3, Section 3 and a paper by Kasoev and Lyamshev [110]) while treating the problem of sound generation in a liquid by radiation (laser) pulses not taking into account the effect of heat conductivity.
2. EXCITATION OF RAYLEIGH WAVE BY RADIATION PULSES As was done in Chapter 6, we can write down corresponding expressions for radial and vertical displacements in the far field of the Rayleigh wave excited by pulses of penetrating radiation,
uR =
(3 − 4 / n 2 )αa 2 I 0 1 − i 16π π cε ρv R v R
R
∞ 1 − iωτ χ
∫
2 2 −∞ 1 + ω τ χ
F (ω ) ×
ω 2τ 2 a exp iω R − t × exp − 4 v R 1 1 + ω τ µ
ω ∗ i ωV1 ( χ 0 ) + ω
2 2 ∗ 1 − vR / cl V2 ( χ 0 ) dω , (7.21)
∞ 1 − iωτ (3 − 4 / n )αa I 0 1 − i χ F (ω ) × uz = 2 2 16π π cε ρv R v R R −∞ 1 + ω τ χ 2
2
∫
ω 2τ 2 a exp iω R − t × exp − 4 v R 1 1 + ω τµ
ω ∗ i ωV2 ( χ 0 ) − ω
2 2 ∗ 1 − vR / cl V1 ( χ 0 ) dω .
RADIATION ACOUSTICS
185
Here τa = a/vR , τµ = (µ/vR) (1 – vR2/cl2)1/2, vR is the propagation velocity of Rayleigh wave, and all other notations correspond to those introduced earlier. If the length of radiation pulse τ is small as against the characteristic times τa and τχ, expressions for displacements take on the form
uR =
(3 − 4 / n 2 )αa 2 I 0 2 AS
∞
ω exp(−ω 2τ a2 / 4)
∫ R (1 + ωτ
8π π cε ρv R v R
0
2 2 µ )(1 + ω τ χ )
×
π π R R − t dω , − t − ωτ χ sin − ω cos − ω 4 vR vR 4 (7.22) uz =
(3 − 4 / n 2 )αa 2 I 0 2 BS 8π π cε ρv R v R
R
∞
ω exp( −ω 2τ a2 / 4)
∫ (1 + ωτ 0
2 2 µ )(1 + ω τ χ )
×
π π R R − t dω , − t + ωτ χ cos − ω sin − ω 4 4 vR vR where 2 2 2 2 ∗ / cl , B = V2∗ ( χ 0 ) + i 1 − v R / cl V1 ( χ 0 ) ; A = iV1∗ ( χ 0 ) + V2∗ ( χ 0 ) 1 − v R
(7.23) ∞
s=
∫ f (t )dt 0
is the “area” of radiation pulse. If n = 2 then A and B are approximately equal to 2. One can see that in the case of short radiation pulses, the shape of sound pulse does not depend on the shape of radiation pulse. If we ignore the effect of heat conductivity, i.e., take τχ = 0 which is always possible when the dimension of a beam of penetrating radiation exceeds 10−5 m, the shapes of sound pulses described by expressions (7.22) are determined only by the quantities τa and τµ. Figure 7.2 gives the shapes of sound pulses for vertical displacements in the case of short radiation pulses and with absence of heat conductivity. One can see that as the depth
186
PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS
of radiation penetration into a substance increases, the amplitude of a sound pulse decreases and its length grows. If the dimension of radiation beam is much smaller than 10−5 m, i.e., τa τµ and (2) τµ = 5τa.
Figure 7.3 Shapes of sound pulses for radial (curve 1) and vertical (curve 2) displacements in the Rayleigh wave in the case of short radiation pulses and strong influence of heat conductivity on the process of sound generation.
3. SOUND GENERATION IN A SOLID HALF-SPACE IN THE PRESENCE OF A LIQUID LAYER AT ITS SURFACE Instead of expression (6.31) characterizing harmonically modulated penetrating radiation, we obtain an analogous expression for a sound pulse generated by a pulse of penetrating radiation in a solid half-space
RADIATION ACOUSTICS
I a2 σ RR ( x, y, z, t ) = 0 8πR
2 2 2 R ω a sin θ exp i t exp ω − − ∫ cl 4cl2 −∞ ∞
αρ µω cos θ 1 exp iω γ − − cl ρC p
187
AF (ω ) ×
G H × 1 − V exp( 2iωγH )
µA exp( − µH ) sin(ωγH ) 2 1 − 1 − (1 − A)(1 − A1 ) exp( −2 µH ) µ 2 + ω 2γ 2 (7.24)
ωγ ( 2 − A1 ) exp( − µH ) cos(ωγH ) + ωγ (1 − A1 ) exp( −2 µH ) − µ 2 + ω 2γ 2 2 (3 − 4 / n 2 )αµ1 1 − iωχ / cl ωA1 exp{−[ µ + (iω / cl ) cos θ ]H } × cε 1 + ω 2 χ 2 / c 4 1 − (1 − A)(1 − A1 ) exp( −2 µH ) l
(1 − V1 )(ω / cl ) cos θ + iµ1 (1 + V1 ) dω , µ12 + (ω 2 / cl2 ) cos2 θ where all notations correspond to the ones introduced in Chapter 6. We have to remember that here, as in Chapter 6, we consider for definiteness only the field of longitudinal waves, and a beam of penetrating radiation is incident vertically at the free surface of a liquid layer. The field of longitudinal waves can be separated unambiguously from the field of transverse waves not only according to the character of polarization but also according to the difference of arrival times of sound pulses. Expression (7.24) is rather complex, and therefore it is interesting to analyze it for different limiting cases. Let exp(−µH) ≈ 1, i.e., absorption of penetrating radiation occurs mainly in the solid half-space. Then, expression (7.24) can be rearranged to the form
188
PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS
2 2 2 I 2 ∞ R ω a sin θ σ RR ( x, y, z, t ) = 0a ∫ exp iω − t exp − 8πR cl 4cl2 −∞ αρ µω cosθ 1 exp iω γ − − cl ρC p
AF (ω ) ×
H ×
2G[ µA1 sin(ωγH ) + 2ωγ ( A1 − 2) sin 2 (ωγH / 2) [1 − V exp(2iωH )]( A + A1 − AA1 )( µ 2 + ω 2γ 2 )
+
(7.25)
2 (3 − 4 / n 2 )αµ1 1 − iωχ / cl × cε 1 + ω 2 χ 2 / c4 l
ωA1 exp[−(iω / cl ) cosθ H ] (1 − V1 )(ω / cl ) cos θ + iµ1 (1 + V1 ) dω . A + A1 − AA1 µ12 + (ω 2 / cl2 ) cos2 θ In this case the layer affects the sound field in the solid half-space mainly because of its wave properties. If the layer thickness H is much smaller than the radius of radiation beam a in this case, the expression for the sound field in the solid half-space gets reduced to a greater extent for not very small observation angles θ (since the product of frequency by the layer thickness can be treated as a small quantity and the exponents containing this product are equal to one),
σ RR ( x, y, z, t ) = ∞
(3 − 4 / n 2 )αµ1 AA1 I0a 2 × 8πcε R A + A1 + AA1
ω 2 a 2 sin 2 θ exp ∫ − 4c 2 l −∞
exp iω R − t F (ω ) × cl
(7.26)
RADIATION ACOUSTICS
1 − iωχ / cl2 ω (1 − V1 )(ω / cl ) cos θ + iµ (1 + V1 ) 1 + ω 2 χ 2 / cl4
µ12 + (ω 2 / cl2 ) cos2 θ
189
dω ,
where the reflection coefficient is reduced now to the form V1 = V′ – DG/(1 + V). If we take A = 1 and the value of ρc for the liquid layer equal to zero in expression (7.26), this expression coincides with the analogous expression (7.8) for the sound field in a solid half-space, and the reflection coefficient V1 is the coefficient of reflection of a plane longitudinal wave from a free boundary of a solid half-space. Let us now consider the opposite limiting case exp (−µH) µ−1, energy is utilized mainly for excitation of longitudinal and Rayleigh waves. And if a > τa, τl, τχ, the roll-off of the spectral density in the integration element in expression (7.35) is determined by the spectrum of radiation pulse F(ω). In this case we may take the exponent in expressions (7.35) to be equal to zero, the parameter τχ equal to one, and the functions Φ1(ωτµ) and Φ2(ωτµ) may be expanded into a series. We restrict ourselves to the first terms of the expansion Φ1(ωτµ) ≈ ωτµΦ1′(0) and Φ2(ωτµ) ≈ Φ2(0). The shape of a sound pulse is determined in this case by the first and second derivatives of the envelope of the penetrating radiation pulse with respect to time,
σ RR ( x, y, z, t ) = −
(3 − 4 / n 2 )αa 2 I 0 4cε R
R M (θ )Φ 2 (0) f ' t − c l
+ (7.36)
R τ l K (θ )Φ1′ (0) f ' ' t − . cl If we consider short radiation pulses, observation is conducted under large angles θ, and the value of heat conductivity is small, i.e., τa >> τ, τµ, τχ, then the roll-off of the spectral density in the integration element in expression (7.35) is determined by the exponential function (−ω2τa2/4). The spectrum of the penetrating radiation pulse can be changed in this case for a constant equal approximately to τ and the functions Φ1(ωτµ) and Φ2(ωτµ) can be expanded into a series as before. In this case we also restrict ourselves to the first terms of the expansion. The parameter τχ is taken to be equal to zero again. An expression for a sound pulse takes on the form
σ RR ( x, y, z, t ) =
(3 − 4 / n 2 )αa 2 I 0 2 π cε Rτ a3
(t − R / c ) 2 l exp − 2 τ a
× (7.37)
2( t − R / c ) 2 R l τ Φ ′ (0) K (θ ) . t − Φ 2 (0) M (θ ) + 1 − 1 µ cl τ a2 In this case the shape of a sound pulse is also determined by the characteristic time τa and does not depend on the particular features of radiation absorption in a substance. In other cases, i.e., when the length of radiation pulse is very small, observation is performed under small angles θ, and under the considerable influence of heat conductivity, the roll-off of the spectral density in the
RADIATION ACOUSTICS
195
integration element in expression (7.35) is determined by the functions Φ1(ωτµ), Φ2(ωτµ), and (1 − iωτχ)/(1 + ω2τχ2). In other words, it is necessary to know a specific form of the function of energy release Q in order to determine the shape of a sound pulse. Some dependences of rather complex functions of energy release are given in Chapter 1, Section 1. In the case when absorption of penetrating radiation in a medium occurs exponentially, the functions Φ1(ωτµ) and Φ2(ωτµ) can be written down in an explicit form Φ1(ωτµ) = ωτµ /(1 + ω2τµ2) and Φ2(ωτµ) = 1/(1 + ω2τµ2). The main features of generated sound fields in the case of such forms of the functions Φ1(ωτµ) and Φ2(ωτµ) have been considered in Section 1 of this chapter.
6. THERMORADIATION GENERATION OF SOUND BY PULSES OF NON-RELATIVISTIC PROTONS We consider sound generation by a non-relativistic proton beam as an example. Lifshits and Pitaevskii considered this case for a liquid [123]. Energy losses by protons for ionization are described by formula (1.1). We should note that the function z / cos ϑ
1−
∫
0
Q dz I 0 cos ϑ
differs from the energy of a proton E only in normalization with respect to the initial energy of a proton E0. Formula (1.1) is true only up to the value of the energy E∗ when a proton captures an electron. If E ≤ E∗, the rate of energy loss decreases sharply and formula (1.1) becomes inapplicable. For example, the value E∗ for water (or ice) is approximate 1 MeV. Thus, if protons with energy 100 MeV are emitted, approximately 99% of energy is released according to formula (1.1). Analysis of this formula shows that energy loss by a non-relativistic proton increases with the decrease of its energy or with distance traveled by it in a substance. If we approximate the dependence of energy loss by an increasing exponential function and break it at the value E = E∗, we can show that Φ1 (ωτ µ ) ≈
2ωτ µ E0 sin(ωτ µ ) − ωτ µ cos(ωτ µ ) + , 2 E∗ 1 + ω 2τ 2 1 + ω 2τ 2 µ
µ
196
PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS
4E m M 4 µ −1 = E02 4π e NZ ln 0 m IM
−1
,
where M and e are the mass and charge of a proton, m is the mass of an electron, N is the number of substance atoms per unit volume, Z is the nuclear charge of substance atoms, and I is the average energy of excitation of substance atoms. Thus, µ−1 ≈ 25 cm at E = 100 MeV in the case of water (and ice). If such a proton beam is incident vertically on the surface of a solid, the pulse length of the proton beam is small, and the observation angle θ is small also, i.e., at τl >> τa, τ, τχ, then a sound pulse represents basically two pulses of positive excessive pressure of length τ in contrast to the pulse of negative excessive pressure in the case of absorption of penetrating radiation under the exponential absorption law in the analogous situation (formula (7.14)):
σ RR ≈ −
(3 − 4 / n 2 )αa 2 I 0 E0 R f t − + τ µ + 8cε Rτ µ E∗ cl (7.38) R f t − − τ µ . cl
Figure 7.5 Shapes of sound pulses generated by (a) a short pulse of a proton beam and (b) a short pulse of laser radiation in the case of small observation angle.
Figure 7.5a presents shapes of sound pulses for short pulses of a proton beam and in the case of small observation angles θ. The shape of a sound pulse generated by penetrating radiation with an exponential law of absorption (e.g., laser radiation) for the same beam parameters (length of
RADIATION ACOUSTICS
197
radiation pulse, beam radius, depth of radiation penetration into a substance, and radiation intensity) as in the case of the proton beam are given in Fig. 7.5b for comparison. We see that for the same small observation angle θ, the shapes of generated sound pulses differ drastically. Only the particular features of different types of penetrating radiation can explain this difference. Thus, the conducted analysis demonstrates that the shape of generated pulses in the far wave field depends essentially on the particular features of absorption of penetrating radiation in a substance only in the case of very short pulses of penetrating radiation and observation of sound pulses under small angles θ. We should note also that when penetrating radiation leads to considerable ionization of a liquid, the velocity of sound propagation in the ionization region can increase essentially [123]. In this case the parameters τa and τl decrease, respectively. Moreover, in the case of substance ionization, the transition of the energy lost by radiation into thermal energy occurs with a certain delay [123]. Then, the time dependence of thermal sources may differ to some extent from the time dependence of a pulse of penetrating radiation, and therefore the envelope of a sound pulse may change also. We have considered sound generation by penetrating radiation in the far wave field. At the same time an analogous situation in the one-dimensional case or in the case of near wave field is very interesting also. In this case the transverse dimensions of a beam of penetrating radiation are large as against the observation distance, and furthermore, this distance is larger than the depth of radiation penetration into a substance. Then, if we consider normal incidence of radiation on the surface of a liquid half-space and the function Q(z, t) = Q(z)f(t), we can write down an expression for sound pressure in the following form (we use the Green function for a one-dimensional reduced wave equation [51]):
p( z, t ) =
αc 4πC p
∞
z exp iω − t F (ω )dω × c −∞
∫
(7.39) ∞
z z Q ( z ) exp − iω exp iω dz . c c −∞
∫
Here c is the sound velocity in a liquid. If the pulse length of penetrating radiation is small compared with the time of wave propagation to the
198
PULSED THERMORADIATION SOURCES OF SOUND IN SOLIDS
observation point, the spectrum F(ω) can be changed for the length of radiation pulse τ and expression (7.39) takes on the form, p ( z, t ) =
αc 2 τ [Q ( z − ct ) − Q ( z + ct )] . 2C p
(7.40)
If we do not take into account the reflected wave −Q(z + ct), the shape of sound pulse corresponds to the shape of the function of energy evolution. This is the basis for the technique of acoustic dosimetry of penetrating radiation (see [26] for example).
Figure 7.6 Dependence of distribution of absorbed energy of electrons in cellophane, which was obtained by an acoustic method according to formula (7.41).
In the case of a solid we can write down in the same way an expression for the component of stress tensor σzz for the normal incidence of short radiation pulses on its surface, small heat conductivity, and approximation of near wave field:
σ zz = −
(3 − 4 / n 2 )αcl2 2cε
τ [Q( z − cl t ) − Q( z + cl t )]
(7.41)
(other components of stress tensor are expressed through the component σzz in the next way: σzx = σzy = σxy = 0 and σxx = σyy = (1 − 2/n2) σzz). Figure 7.6 [25] gives the dependence of distribution of absorbed energy of electrons in cellophane, which has been obtained on the basis of analysis of an acoustic signal according to formula (7.41). In conclusion of this section we should note the following. We have considered here the influence of the particular features of absorption of penetrating radiation on sound generation in a liquid or solid half-space (in the case of a solid half-space we treated only waves in the bulk of the medium). While considering other specific problems of sound generation by
RADIATION ACOUSTICS
199
penetrating radiation (e.g., surface waves, normal modes in waveguides, waves in layered media, etc.), one may use the same solution technique that has been described here. However, if we have already some specific solution for a certain law of penetrating radiation absorption, and the characteristic scale of a problem is greater than the depth of radiation penetration into a substance l = µ−1, we can obtain a certain “zero” approximation of solution of a new problem using a new value of l = µ−1 in the old solution.
CHAPTER 8
Moving Thermoradiation Sources of Sound A moving thermoradiation source of sound arises in the process of motion of a beam of penetrating radiation along the surface of a condensed medium. Theoretical studies of sound field of such sources were performed apparently for the first time by Bozhkov, Bunkin, and Kolomenskii [31, 32, 34, 112, 113], Esipov [95], Lugovoi and Strel’tsov [124], and Lyamshev and Sedov [136, 148]. These authors considered sound excitation by a moving laser beam with modulated light intensity. Important features of thermoradiation generation of sound (as compared with traditional techniques) are simple realization of a sound source moving with almost any velocity and acceleration due to the absence of medium resistance, an opportunity for smooth and continuous tuning of modulation frequency of intensity of penetrating radiation (in particular, by changing of the velocity of beam motion, i.e., on account of the Doppler effect), and the absence of side-lobes in the directivity pattern. This chapter considers particular features of generation of a sound field created by an intensity-modulated beam of penetrating radiation scanning the surface of a condensed medium.
201
202
MOVING THERMORADIATION SOURCES OF SOUND
1. SOUND GENERATION BY A MOVING THERMORADIATION PULSED SOURCE IN A LIQUID We consider here the particular features of sound generation in a liquid by a beam of penetrating radiation with intensity modulated by pulses of arbitrary shape, which moves along the liquid surface. We impose almost no restrictions upon the velocity of beam motion along the liquid surface and the shape of its trajectory. We assume only that the trajectory of beam motion is located in a finite area of liquid surface and the sound field is treated in the far wave field with respect to the dimensions of this area. We will obtain general expressions characterizing the spectrum of the sound field of a moving thermoradiation source of sound. The next cases are studied in detail: (a) uniform motion of a beam along a finite trajectory; (b) oscillatory motion of a beam; and (c) uniform motion of a beam along a circle. An interesting particular feature is revealed: in the case of uniform and rectilinear motion of a beam of penetrating radiation, sound generation occurs in the same way as in the case of a motionless beam but at a certain effective shape of a radiation pulse. In this case the envelope of the sound signal in the Cherenkov direction does not depend on the shape of sound pulse and is determined by the geometrical parameters of the region of effective heat release and the observation angle, while the amplitude of the acoustic signal is directly proportional to the energy of the pulse of penetrating radiation. Let us analyze the particular features of the spectrum of generated acoustic signal for the cases of an oscillating beam of penetrating radiation and a beam moving along a circle, investigate in detail the quasi-monochromatic mode of sound generation by such sources, and compare the characteristics of these sources to the characteristics of a motionless quasi-monochromatic radiation-acoustic radiator. Let a beam of penetrating radiation be incident in the positive direction of the axis z at the surface of a half-space filled with a liquid. The function f(t) describes the shape of a radiation pulse (we assume that this function is bounded and continuous) and the transverse distribution of energy in the beam has the form I(x, y) = I0 exp [− (x2 + y2)/a], where a is the effective radius of the beam section at the liquid surface. We assume that the radiation spot moves along the liquid surface along a trajectory with the coordinates x0(t), y0(t). Then the power density of thermal sound sources in the liquid is determined by the expression, Q ( x, y, z, t ) = AµI 0 f (t ) exp[−( x − x0 (t )) 2 / a 2 − ( y − y0 (t )) / a 2 − µz ] .
RADIATION ACOUSTICS
203
An equation for the spectrum of sound pressure generated by a moving beam of penetrating radiation in the liquid has the form
∆pω + k 0 pω = −
AµI 0α exp(− µz ) Cp
∞
2 iωt − [ x − x0 (t )] − exp ∫ a2 −∞
(
[ y − y0 (t )]2 f (t ) f ′(t ) + 2 2 xx0′ (t ) − x0 (t ) x0′ (t ) + 2 a a
(8.1)
yy0′ (t ) − y0 (t ) y0′ (t ) , where k0 is the complex wave number of sound and f ′0(t), x′0(t), and y′0(t) are the derivatives of the functions f0(t), x0(t), and y0(t). Let the trajectory of motion of a penetrating radiation beam be located in a limited area of the liquid surface. We are interested in the sound field in the far wave zone1. We obtain an expression for the spectrum of sound pressure in the far wave zone according to Kasoev and Lyamshev [109, 110] (see Chapter 5, Section 2):
pω = −
2 exp(ikr0 ) ω τ µ α × AI 0 a 2 2r0 1 + ω 2τ 2 Cp µ
(8.2) ω 2τ 2 a exp − 4
− q(ω )r F ∗ (ω ) , 0
where F ∗ (ω ) =
∞
∫ f (t ) exp(iωt + ik sin θ [ x0 (t ) cos ϕ + y0 (t ) sin ϕ ])dt ,
(8.3)
−∞
τµ = (cos θ)/µc, τa = (a sin θ)/c , c is the sound velocity, θ is the angle between the axis z and the radius-vector of the observation point r0, ϕ is the 1
We mean the Fraunhofer zone with respect to the area of motion of radiation spot along the liquid surface and the upper limiting frequency of the spectrum of sound signal taking into account sound attenuation in a liquid.
MOVING THERMORADIATION SOURCES OF SOUND
204
angle between the axis x and the projection of r0 onto the plane x, y; q(ω) = Im k0 is the coefficient of sound attenuation in the liquid; k = Re k0. If we compare formula (8.2) to the analogous expression for the spectrum of sound pressure of a motionless pulsed radiation-acoustic source, i.e., formula (5.2) (see Chapter 5 and also [88, 151]), we can see that these formulae almost coincide and differ only in the fact that the spectral density of laser pulse F(ω) in formula (8.2) is changed for the ∗ function F (ω). The last depends not only on the shape and length of a radiation pulse but also on the characteristics of motion of a radiation spot (a beam of penetrating radiation) along the liquid surface. Expression (8.2) is obtained on the grounds of rather general assumptions indicated above. It follows from this expression that the spectrum of sound pressure in a liquid is determined by the geometrical parameters of the region of absorption of penetrating radiation in a liquid as ∗ in the case of a motionless beam on the one hand and by the function F (ω), which depends on the spectrum of radiation pulse and motion parameters of penetrating radiation beam on the other hand. An analytic representation of the spectrum of sound field in a liquid (expression (8.2)) is very convenient for consideration of various particular cases of sound radiation by a moving pulsed thermoradiation sound source. We consider some of them below. Let a radiation spot be moving uniformly and rectilinearly with velocity ∗ V along the axis x, i.e., x0(t) = Vt, y0(t) = 0. Then the function F (ω) can be expressed with the help of the spectrum of radiation pulse in a simple way: F ∗ (ω ) = F [(1 − β ∗ )ω ] , ∗
(8.4)
where β = (V/c) sin θ cos ϕ. One can see from formulae (8.2) and (8.4) that sound generation by a moving uniformly and rectilinearly pulsed radiation-acoustic source occurs ∗ in the same way as by a motionless but “compressed” by |1 − β | times ∗ ∗ radiation pulse, which is described by the function f[t/(1 − β )]/|1 − β | and ∗ has the effective length |1 − β |τ, where τ is the length of a penetrating ∗ radiation pulse. We should note that if 1 − β < 0, the “effective” pulse ∗ ∗ f[t/(1 − β )]/|1 − β | is not only compressed but also inverted in time with respect to the radiation pulse. This is connected with the fact that in the case of supersonic motion of a radiation-acoustic source, sound disturbances, which have been produced by the source later, arrive at the observation points in certain directions sooner. Thus, almost all discussion and results of Chapter 5, Section 3 (see also [110]) can be transferred directly to the case of a moving source if we ∗ ∗ consider the effective radiation pulse f[t/(1 − β )]/|1 − β | with length |1 − ∗ β |τ. Therefore, we do not consider the details discussed in Chapter 5 but
RADIATION ACOUSTICS
205
only the particular features of sound radiation in the Cherenkov direction. We mean the direction where sound disturbances from various points get added in-phase. The Cherenkov direction is determined by the equation
β ∗ = (V / c) sin θ cos ϕ = 1 . If this condition is satisfied then, as it follows from expression (8.4), the * function F (ω) takes on an especially simple form F ∗ (ω ) = F (0) =
∞
∫ f (t )dt = σ ,
−∞ *
i.e., F (ω) does not depend on frequency and is equal to the “area” of the * radiation pulse σ. Substituting F (ω) = σ into expression (8.2) and performing the inverse Fourier transformation, we obtain an expression describing a sound field in the Cherenkov direction: p=−
γ2 AI 0 a 2ασ 4 exp − s2 8C p r0τ µ2 π s
2 − exp s × 4
(8.5)
(exp(−γ ) Erfc ( s / 2 − γ / s ) + exp(γ ) Erfc ( s / 2 + γ / s )) ,
where γ = (r0/c − t)/τµ, s = (τa2 +4Cr0)1/2/τµ, and Erfc z is the complementary error function. In order to simplify calculation, we take here q(ω) = Cω2, where C is a certain constant. We note that expression (8.5) coincides with the formula obtained in the process of consideration of sound generation by a very short radiation (laser) pulse [109]. As follows from analysis of expression (8.5), the shape of the envelope of a sound pulse in the Cherenkov direction depends neither on the shape nor on the length of penetrating radiation pulse. The envelope of a sound pulse is determined by the geometrical parameters of the region of effective heat release a, 1/µ, the parameter Cr0, and the observation angle θ, while the pulse amplitude is directly proportional to the energy of penetrating radiation pulse πa2I0σ. The shape of the envelope of a sound pulse in the Cherenkov direction calculated according to formula (8.5) is given by curve 1 in Fig. 8.1. Here the vertical axis presents the amplitude of a sound pulse normalized to the
206
MOVING THERMORADIATION SOURCES OF SOUND
value Aa2I0τα/(4r0τµ2Cp) and the horizontal axis presents the quantity γ = (r0/c − t) / τ, i.e., the dimensionless time. We have assumed in the process of calculation that s = 2. It is necessary to note that the shape of a sound pulse in these coordinates is determined by the single parameter s = (τa2 + 4Cr0)1/2/τµ . If the parameter s decreases, the pulse width (in the coordinate γ) decreases and the amplitude grows.
Figure 8.1 (1) Envelope of a sound pulse in the Cherenkov direction and (2, 3) shapes of pulses of “switching-on” and “switching-off”.
Some particular features of sound generation by radiation pulses of arbitrary shape in the process of uniform and rectilinear motion of a beam of penetrating radiation along a liquid surface have been considered above. It would be interesting to consider in more detail the case of intra-pulse quasi-monochromatic modulation of penetrating radiation in intensity. For example, let 1 − cos ω 0 t , | t | ≤ τ , f (t ) = | t | >τ . 0 , Let us take ω0τ = 2πn (where n is a certain natural number) in order for the function f(t) to be continuous. The spectrum of the function f(t) consists of three narrow (of width of the order of magnitude of 1/τ) bands at zero and at the frequencies ±ω0:
RADIATION ACOUSTICS
F (ω ) = 2
sin(ωt )
ω
−
207
sin[(ω + ω 0 )τ ] sin[(ω − ω 0 )τ ] . − ω + ω0 ω − ω0
*
We obtain an expression for F (ω), F ∗ (ω ) = 2
sin[(1 − β ∗ )ωt ] (1 − β ∗ )ω
−
sin{[(1 − β ∗ )ω + ω 0 ]τ } (1 − β ∗ )ω + ω 0
− (8.6)
sin{[(1 − β ∗ )ω − ω 0 ]τ } . (1 − β ∗ )ω − ω 0 Proceeding from expression (8.2) we can determine that the form of the spectrum of sound signal pω generated by penetrating radiation is * determined not only by the function F (ω) but also by the product of the 2 2 2 rational ω τµ/(1 + ω τµ ) and exponential exp [−ω2(τa2/4 + Cr0] functions. For example, let the spectrum of this product be determined mainly by the exponential function and therefore, be limited by the frequency of the order of magnitude of 2(τa2 + 4Cr0)−1/2. In this case the spectrum of the functions * F (ω) represents three bands at zero and at the Doppler frequencies * * ±ω0/|1 − β | . Let us consider two cases: ω0 > 2|1 − β |(τa2 + 4Cr0)−1/2 and * −1/2 2 ω0 < 2|1 − β |(τa + 4Cr0) . * In the first case, when ω0 > 2|1 − β |(τa2 + 4Cr0)−1/2 and in particular in * observation directions close to the Cherenkov direction, where |1 − β | 2π/Ω), then, as follows from expressions (8.2) and (8.9), the spectrum of sound signal consists of two bands with width of the order of magnitude of 1/τ at the frequencies nΩ multiple to the frequency of beam motion, where n is an integer number. In the opposite case when the pulse length is smaller than the motion period, the spectrum of sound signal is continuous. Finally, in the limiting case of a very long pulse of penetrating radiation (τ >> 2π/Ω), the width of the bands at the frequencies nΩ becomes very small and considering them approximately very narrow, we obtain an expression determining sound pressure in a liquid
(
)
2 α AI 0 a 2 ∞ n ω nτ µ p=− ∑ (−1) 1 + ω 2τ 2 exp − ω n2τ a2 / 4 − q(ω n )r0 × C p r0 n µ n =1
210
MOVING THERMORADIATION SOURCES OF SOUND
× J n (k n b sin θ cos ϕ ) cos[ω n (t − r0 / c)] ,
(8.10)
where ωn = nΩ and kn = ωn/c. It is interesting to compare the characteristics of sound radiation by an “oscillating” radiation-acoustic source at the lower frequency ωl = Ω for example with those of a motionless monochromatic radiation-acoustic source. One can see that the directivity pattern of the source performing oscillatory motion with frequency Ω differs from the directivity pattern of the motionless source by the factor 2J1(k1b sin θ cos ϕ); and, in the selected direction in the case of the corresponding length of the beam track at the surface (so that k1b sin θ ≈ 1.8), the pressure amplitude in the plane of motion exceeds the sound amplitude from a motionless monochromatic radiation-acoustic source approximately by 20%. It is necessary to note also that k1b = Ωb/c = Vm/c, where Vm is the maximum speed of motion of an oscillating source, and since k1b > 1, it radiates sound effectively when the maximum speed of motion is larger than the sound velocity in the liquid. Moreover, differing from a motionless monochromatic radiation-acoustic source, the sound field of a source performing oscillatory motion is concentrated close to the motion plane and it is absent in the plane perpendicular to the motion direction. Figure 8.2 presents the surfaces of the functions
p1 (θ , ϕ ) =
2 Ω 2τ 2 α AI 0 a 2 Ω τ µ a − q (Ω) r × exp − 0 C p r0 1 + Ω 2τ 2 4 µ
Ω J1 b sin θ cos ϕ , c which describe the angular dependence of the amplitude of sound pressure produced by a beam of penetrating radiation performing oscillatory motion at the frequency Ω. The amplitude of sound pressure measured in relative units is plotted in the vertical axis, while the horizontal axes present the values of observation angles θ and ϕ from 0 to 90°. Calculation was performed for the next values of the problem parameters: k1a = 2, k1/µ = 21/2, q(Ω) = 0, i.e., not taking into account sound attenuation in a liquid that is unimportant in this case, k1b = 3 (Fig. 8.2a), k1b = 6 (Fig. 8.2b), and k1b = 9 (Fig. 8.2c). The relief of the surface of the function p1(θ, ϕ) is plotted by the curves representing sections of the p1(θ, ϕ) by the planes θ = const and ϕ = const. The sections are made per three degrees.
RADIATION ACOUSTICS
211
Figure 8.2. Angular dependences of sound pressure.
One can see in the figure the major features of sound radiation by an oscillating radiation-acoustic source. The maximum of the directivity pattern is located in the plane of beam motion ϕ = 0. There is no sound field in the plane perpendicular to the direction of motion ϕ = 90°. Changing the amplitude of beam motion b, one can change the direction where sound radiation is maximal or create “gaps” in the directivity pattern. In conclusion let us consider the particular features of sound radiation by a radiation-acoustic source in the case of motion of a radiation spot along a circle: x0(t) = B sin Ωt, y0(t) = B cos Ωt. In this case F ∗ (ω ) =
∞
∑ J n (kb sin θ ) exp(inϕ ) F (ω + nΩ) .
n = −∞
(8.11)
In the case of a very long radiation pulse τ >> 2π/Ω, its spectrum can be written down approximately in the form F (ω + nΩ) = 2πδ (ω + nΩ) . (8.12) Substituting this expression into formula (8.11) and then into expression (8.2) and performing the inverse Fourier transformation, we obtain an expression for sound pressure
p=−
2 2 2 α AI 0 a 2 ∞ n ω n τ µ exp − ω n τ a − q(ω )r × ( ) 1 − n 0 ∑ C p r0 4 1 + ω n2τ µ2 n =1
212
MOVING THERMORADIATION SOURCES OF SOUND
× J n (k n B sin θ ) cos[ω n (t − r0 / c) + nϕ ] .
(8.13)
As one would expect by proceeding from general physical concepts, the directivity pattern of a radiation-acoustic source performing motion along a circle at a certain frequency is the product of expressions describing the directivity pattern of a motionless monochromatic radiation-acoustic source and the directivity pattern of sound radiation by a body moving along a circle [88]. In this case as in the previous one, the amplitude of sound pressure at the frequency Ω can be larger than the amplitude of sound pressure in the case of a motionless radiation-acoustic source if a certain value of circle radius B is selected. In this case estimations show (we do not give them here) that the efficiency of sound generation by a beam of penetrating radiation moving along a circle, which is determined by the ratio of the power of generated sound oscillations to the power of penetrating radiation, is almost the same as in the case of a motionless monochromatic radiation-acoustic source. The results obtained above can be extended readily to the case when a beam performs motion along a circle and the intensity of penetrating radiation is modulated monochromatically with sound frequency ω0. If we assume that the length of radiation pulse is large as against both the modulation period (τ >> 2π/ω0) and the period of beam rotation (τ >> 2π/Ω), i.e., if we consider the quasi-monochromatic mode of sound generation, it is possible to demonstrate that sound waves with frequencies ωn± = ω0 ± nΩ, where n = 0, 1, 2, …, are emitted apart from waves at the frequencies multiple to the rotation frequency nΩ with the characteristics of acoustic radiation described by expression (8.13). Directivity patterns of sound sources at each of these frequencies are described by the expression
Pn ± =
2 ω2 τ 2 α AI 0 a 2 m ω n ±τ µ exp − n ± a − q (ω n ± )r0 × Cp 2r0 1 + ω 2 τ 2 4 n± µ
(8.14) J n ( k n ± B sin θ ) , where m is the modulation index. Summarizing, we note the next particular features of sound generation in a liquid by a beam of penetrating radiation moving at its surface. In the case of a uniform rectilinear motion of a beam operating in the pulse mode, the characteristics of generated sound signal are the same as in the case of a motionless beam but for a certain effective shape of radiation pulse. A sound signal with the shape of envelope independent of the shape
RADIATION ACOUSTICS
213
of radiation pulse, and determined by the parameters of the region of effective heat release and the observation angle, is generated in the Cherenkov direction. The amplitude of this pulse is directly proportional to the energy of penetrating radiation pulse. In the particular case of intrapulse quasi-monochromatic modulation of penetrating radiation, the presence of modulation does not influence the characteristics of the sound field in the Cherenkov direction and the directions of observation close to it. The quasi-monochromatic component at the Doppler frequency arises in the sound field only at a certain angular distance from the Cherenkov direction. In the case of oscillatory motion or rotation of a beam of penetrating radiation, and in the case of a long enough radiation pulse, only sound disturbances at frequencies multiple to the frequency of oscillatory motion or rotation frequency are generated. In the case of a certain set of the parameters of beam motion, the efficiency of sound generation by an oscillating or rotating radiation beam is not smaller than the efficiency of sound generation by a motionless quasi-monochromatic radiation-acoustic source, but the sound beam is very narrow. We should note also that the analytical presentation of the spectrum of sound field in a liquid (8.2) can be useful for consideration of the particular features of thermoradiation generation of sound in a liquid in the case of other forms of the trajectories of motion of a beam of penetrating radiation at the surface of a liquid.
2. SOUND EXCITATION BY A MOVING THERMORADIATION PULSED SOURCE IN SOLIDS Now let us consider sound excitation by a pulsed moving thermoradiation source in solids. Let a beam of penetrating radiation be incident in the positive direction of the axis z on the boundary of a solid homogeneous and isotropic halfspace z > 0. We take the Gaussian intensity distribution in the beam as usual. Let us assume that the radiation spot moves at the solid surface along a trajectory with the coordinates x0(t), y0(t). We assume also that the power density of sound sources Q in a solid is determined by the expression Q ( x, y, z, t ) = µI 0 f (t ) exp[−( x − x0 (t )) 2 / a 2 − (8.15) ( y − y0 (t )) 2 / a 2 − µz ] ,
MOVING THERMORADIATION SOURCES OF SOUND
214
where f(t) describes the time shape of the pulse of penetrating radiation, and the coefficient of transmission through the solid boundary is taken into account directly in the expression for the radiation intensity I(x, y). Let us write down an expression for the Fourier transforms of the scalar potential Φω and the only (because of the axial symmetry of the problem) component of the vector potential Ψω of the displacement vector in the sound field: 2 (3 − 4 / n 2 )α iω iω 3 2 ω ∆ + ∆ + ∆ + Φω = − µI 0 exp(− µz ) F ∗ (ω ) , 2 2 χ χ c ρ ε cl χcl (8.16) 2 ∆ + ω Ψ = 0 , ω ct2 where F ∗ (ω ) =
∞
2 2 iωt − ( x − x0 (t )) − ( y − y 0 (t )) ( ) exp f t ∫ a2 a2 −∞
dt .
We are interested in determining the sound field in the far wave zone, i.e., the Fraunhofer zone with respect to the dimensions of the area of motion of penetrating radiation beam and the upper frequency limit of the spectrum of sound signal. We obtain expressions for the components of stress tensor σRR and σRθ originating from longitudinal and transverse waves, respectively and for vertical displacements uz in the Rayleigh wave at the boundary of a solid just in the same way as above (see previous sections and also [151, 159]):
σ RR =
(3 − 4 / n 2 )αa 2 I 0 8πcε R
∞
∫
−∞
ω exp(−ω 2τ a2 / 4) exp[iω ( R / cl − t )] (1 + ω 2τ χ2 )(1 + ω 2τ µ2 )
×
(1 − iωτ χ )[ωτ µ K (θ ) + iM (θ )]F∗l (ω )dω ,
σ Rθ =
(3 − 4 / n 2 )αa 2 I 0 8πcε R
∞
∫
−∞
ω exp(−ω 2 n 2τ a2 / 4) exp[iω ( R / ct − t )] (1 + ω 2τ χ2 )(1 + ω 2τν2 )
×
RADIATION ACOUSTICS
215
(1 − iωτ χ )(i − ωτν )V2 (θ ) F∗t (ω )dω ,
uz =
(3 − 4 / n 2 )αa 2 I 0 1 − i 16cε ρ (πv R ) 3 / 2
R
∞
∫
2 exp(−ω 2 a 2 / 4v R ) exp[iω ( R / v R − t )]
2 2 2 2 1/ 2 − ∞ (1 + ω τ χ )[1 + ( ω / µv R )(1 − v R / cl )]
∗ − ω 1 − ω i V 2 ω
τa =
(8.17)
1/ 2
2 vR cl2
×
V1∗ F∗R (ω )dω ,
a sin θ cosθ (1 / n 2 − sin 2 θ )1 / 2 χ , τµ = , τν = , τχ = , cl µcl µct cl2
R is the distance from the region of sound generation to the observation point, K(θ) = 1 − V1(θ), M(θ) = 1 + V1(θ), θ is the angle between the axis z and the direction to the observation point, ϕ is the angle between the axis x and the projection of R onto the plane (x, y), and vR is the velocity of Rayleigh wave, ∞
F∗l =
ω f (t ) exp iωt + i sin θ [ x0 (t ) cos ϕ + y 0 (t ) sin ϕ ] dt , c l −∞
∫
∞
F∗t =
ω f (t ) exp iωt + i sin θ [ x0 (t ) cos ϕ + y 0 (t ) sin ϕ ] dt , ct −∞
∫
(8.18)
∞
F∗R =
ω f (t ) exp iωt + i sin θ [ x0 (t ) cos ϕ + y0 (t ) sin ϕ ] dt . vR −∞
∫
The coefficients of reflection of longitudinal and transverse waves V1(θ) and V2(θ) from the free boundary of a solid in the case of incidence of a longitudinal wave at the boundary are expressed as follows: V1 (θ ) =
2 sin θ sin 2θ (n 2 − sin 2 θ )1 / 2 − (n 2 − 2 sin 2 θ ) 2 2 sin θ sin 2θ (n 2 − sin 2 θ )1 / 2 + (n 2 − 2 sin 2 θ ) 2
,
216
MOVING THERMORADIATION SOURCES OF SOUND
V2 (θ ) = −
4n sin θ cos 2θ (1 − n 2 sin 2 θ )1 / 2 2
2
1/ 2
2 sin θ sin 2θ (1 − n sin θ )
2
+ n cos 2θ
,
(8.19)
*
and the analogs of reflection coefficients in the case of Rayleigh wave V1 * and V2 have the form
i vR V1∗ = − 2 L cl
2 2 2cl n − 2 vR
1/ 2
c2 L = l − n2 v2 R
2 1/ 2 2 2 n cl 2 2cl ∗ , V2 = L 2 − 1 n − 2 , v R v R
2 2 3c 2 (c 2 / v R − 1)cl2 / v R + 2 − l − l 2 2 2 2 vR cl / v R − n
(8.20)
c2 2c 2 2 l − 1 l − n 2 . 2 2 vR vR If we compare expressions (8.17) with analogous expressions (7.3) (see Chapter 7), we can see that these expressions differ only in the fact that in expressions (8.17) the spectral density of the pulse of penetrating radiation ∞
F (ω) =
∫ f (t ) exp(iωt )dt
−∞
is changed for corresponding expressions (8.18). The last depend not only on the shape and length of a pulse of penetrating radiation but also on the characteristics of motion of a radiation beam at the solid surface. Expressions (8.17) are obtained on the basis of rather general assumptions. It follows from them that in the general case, sound fields in a solid are determined by the geometrical parameters of the region of radiation absorption, as in the case of a motionless source on the one hand, and by the functions F∗l, F∗t, and F∗R depending on the spectrum of a pulse of penetrating radiation and the parameters of beam motion on the other hand. Analytical expressions obtained for sound fields are convenient for consideration of various particular cases of motion of radiation-acoustic sources of sound. Let us consider some of them.
RADIATION ACOUSTICS
217
Let a radiation spot be moving uniformly and rectilinearly with the velocity v along the axis x, i.e., x0(t) = vt, y0(t) = 0. In this case the functions F∗ are expressed in a simple way with the help of the spectrum of an optical pulse F(ω): F∗l (ω ) = F [(1 − β l∗ )ω ] , F∗t (ω ) = F [(1 − β t2 )ω ] , (8.21) ∗ )ω ] , F∗ R (ω ) = F [(1 − β R ∗
∗
∗
where βl = (v/cl) sin θ cos ϕ, βt = (v/ct) sin θ cos ϕ , and βR = (v/cR) sin θ cos ϕ . It follows from expressions (8.17) and (8.21) that sound generation by a moving uniformly and rectilinearly pulsed radiation-acoustic source occurs ∗ in the same way as by a motionless source, which is compressed by |1 − β | ∗ ∗ times, described by the function f[t/(1 − β )]/|1 − β |, and has the effective ∗ ∗ length |1 − β |τ. Here τ is the length of penetrating radiation pulse and β are the corresponding Mach numbers for longitudinal, transverse, and ∗ Rayleigh waves. It is necessary to note that if 1 − β < 0 for some type of ∗ ∗ waves, the “effective” pulse f[t/(1 − β )]/|1 − β | is not only compressed but also inverted in time with respect to the pulse of optical radiation. As has been noted above, this is connected with the fact that in the process of supersonic motion of a source, sound disturbances produced by a radiationacoustic source later arrive at the observation points in certain directions earlier. In this case a situation may arise that motion may be subsonic for some types of waves, e.g., longitudinal waves, but supersonic for other waves, e.g., Rayleigh waves. Thus, almost all considerations in Section 1 of this chapter and the results by Lyamshev and Chelnokov [151, 159] can be applied directly to the case of a moving thermoradiation source if we consider an effective ∗ ∗ ∗ pulse of penetrating radiation f[t/(1 − β )]/|1 − β | with length |1 − β |τ. Therefore, we do not give all details here. They are covered in the section devoted to sound generation by radiation pulses in a solid. Here we consider only the particular features of sound generation in the Cherenkov directions, meaning the directions where sound disturbances from different points of motion trajectory of a radiation-acoustic source are added inphase. The Cherenkov directions for different types of waves are determined by equations v v v sin θ cos ϕ = 1 , sin θ cos ϕ = 1 , cos ϕ = 1 . cl ct vR
218
MOVING THERMORADIATION SOURCES OF SOUND
If these conditions are satisfied, then it follows from expressions (8.21) that the functions F∗l, F∗t, and F∗R acquire an especially simple form ∞
F∗l (ω ) = F∗t (ω ) = F∗ R (ω ) = F (0) =
∫ f (t )dt = s ,
−∞
i.e., F∗l, F∗t, and F∗R do not depend on frequency and are equal to the “area” s of the radiation pulse. Substituting F∗(ω) in expressions (8.17), we obtain expressions describing sound fields of longitudinal and Rayleigh waves in the Cherenkov directions,
σ RR =
2 2 τµ (3 − 4 / n 2 )αa 2 I 0 s exp(τ a / 4τ χ ) K (θ ) × M (θ ) + 2 2 8cε R τχ τ µ − τ χ
t − R / cl exp τχ
Erfc τ a + t − R / cl 2τ χ τa
τχ exp(τ a2 / 4τ µ2 ) V1 (θ )1 − 2 2 τµ τ µ − τ χ τ t − R / cl Erfc a − 2τ µ τa
τχ − 1 + τµ
exp − t − R / cl τµ
exp t − R / cl τµ
0
, (8.22)
1/ 2
ω 1 / 2 exp( −ω 2 a 2 / 4v R2 )
∫ [1 + (ω / µv
×
Erfc τ a + t − R / cl 2τ µ τa
2 (3 − 4 / n 2 )αa 2 I 0 s 2 ∗ v R uz = V1 + i1 − 8cε ρ (πv R ) 3 / 2 R1 / 2 cl2 ∞
+
2 2 1/ 2 ](1 + ω 2τ χ2 ) R )(1 − v R / cl )
V1∗ ×
×
π π R R − t dω , − t + ωτ χ cos − ω sin − ω 4 vR vR 4
RADIATION ACOUSTICS
219
where ∞
2
Erfc ( x) = (2 / π ) ∫ e − t dt x
is the complementary error function. It follows from the analysis of expressions (8.22) that the shapes of the envelopes of sound signals in the Cherenkov directions depend on neither the shape, nor the length of a radiation pulse. Envelopes of sound pulses are determined by the geometrical parameters of the region of effective heat release a and 1/µ and the observation angle θ, while the amplitudes of pulses are directly proportional to the energy of a pulse of penetrating radiation πa2I0s. If a pulse of penetrating radiation is modulated quasimonochromatically in intensity, for example in the form 1 − cos ω 0τ , | t | ≤ τ , f (t ) = | t | >τ , 0 , where ω0τ = 2πk and k is a natural number, then analogously to the way it has been done in the previous section of this chapter for the case of sound generation in a liquid, we can obtain similar expressions for a solid. Analysis of these expressions allows us to note the following features of sound fields generated by moving uniformly and rectilinearly thermoradiation sources of sound in the case of intra-pulse quasimonochromatic modulation of penetrating radiation in intensity. The shapes of the envelopes of generated sound signals in the Cherenkov directions do not depend on the shape of penetrating radiation pulse. The envelopes of sound signals are determined by the geometrical parameters of the region of effective heat release a and 1/µ and the observation angle θ, while the amplitudes are directly proportional to the energy of radiation pulse. In the observation directions close to the Cherenkov directions, where ∗ ∗ |1 − β | 2|1 − β /τa|, sound fields represent a pair of pulses. These pulses are the responses to “switching-on” and “switching-off” a thermoradiation source. If we move away from the Cherenkov directions, the time interval between the pulses of “switchingon” and “switching-off” increases (this interval is of the order of magnitude ∗ ∗ of 2|1 − β |τ). If the conditions ω0 < 2|1 − β |/τa are satisfied, an almost ∗ sinusoidal filling at the Doppler frequency ω0/|1 − β | arises between these pulses.
220
MOVING THERMORADIATION SOURCES OF SOUND
Now let us consider the case of oscillatory motion of a radiation spot at the surface of a solid. Let a beam of penetrating radiation perform oscillatory motion along the axis x according to the law x0(t) = b sin Ωt, y0(t) = 0. We obtain for F∗(ω): F∗l (ω ) =
F∗t (ω ) =
∞
ω J m b cos ϕ sin θ F (ω + mΩ) , cl m = −∞
∑ ∞
ω J m b cos ϕ sin θ F (ω + mΩ) , ct m = −∞
F∗ R (ω ) =
∑
(8.23)
∞
ω J m b cos ϕ F (ω + mΩ) , vR m = −∞
∑
where Jm is the Bessel function. If the length of radiation pulse is larger than the period of beam motion τ > 2π/Ω, then as follows from expressions (8.17) and (8.23), the spectrum of sound signals consists of bands with width of the order of magnitude of 1/τ at frequencies multiple to the frequency of beam motion mΩ, where m is an integer number. In the opposite case when the length of radiation pulse is smaller than the motion period, the spectra of sound signals are continuous. Finally, in the limiting case of a very long radiation pulse, i.e., when τ >> 2π/Ω, the width of bands at the frequencies mΩ becomes very small and assuming them infinitely narrow, we obtain expressions determining sound fields in a solid,
σ RR =
2 τ 2 / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m a (−1) m × ∑ 2 2 2τ 2 ) 2cε R ( 1 )( 1 ω τ ω + + m χ m µ m =1
2 ω τ µ K (θ ) cos[ω m (t − R / cl )] − J m m b sin θ cos ϕ {ω m c l 3 2 ωm τ χ τ µ K (θ ) sin[ω m (t − R / cl )] + ω m τ χ M (θ ) cos[ω m (t − R / cl )] +
ω m M (θ ) sin[ω m (t − R / cl )]} ,
RADIATION ACOUSTICS
σ Rθ =
221
2 2τ 2 n a / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m × (−1) m ∑ 2 2 2 2 2cε R (1 + ω mτ χ )(1 + ω m τν ) m =1
2 ω J m m b sin θ cos ϕ {ω m cos[ω m (t − R / ct )]V2 (θ )(τ χ − τν ) + ct 2 ω m sin[ω m (t − R / ct )]V2 (θ )(1 + ω m τ χ τν )} ,
(8.24) 2 2 1/ 2 ∗ ∗ (3 − 4 / n 2 )αa 2 I 0 V2 + i (1 − v R / cl ) V1 uz = × R π 3 / 2 cε ρ (2v R ) 3 / 2 ∞
∑
m =1
( −1) m
2 2 2 ω m exp( −ω m a / 4v R ) 2 2 (1 + ω m τ χ )[1 + ω m (1 − v R2 / cl2 )]1 / 2 / µv R
×
π π R R − t × − t + ω mτ χ cos − ω m sin − ω m 4 4 vR vR ω J m m b cos ϕ , vR where ω = mΩ. We can see that the directivity patterns of a radiation-acoustic source performing oscillatory motion at the frequency Ω differ from the directivity patterns of a motionless source by the factors 2Jm[(ω/cl)b sin θ cos ϕ], 2Jm[(ω/ct)b sin θ cos ϕ], and 2Jm[(ω/vR)b cos ϕ], respectively, and in the chosen direction in the case of a certain selection of the track of radiation beam at the surface (so that either b(ω1/cl) sin θ ≈ 1.8, or b(ω1/ct) sin θ ≈ 1.8, or b(ω1/vR) sin θ ≈ 1.8) the amplitudes of corresponding sound fields in the motion plane exceed the amplitudes of sound fields from a motionless monochromatic radiation source by 20% approximately. We should note also that (ω1/cl)b = vmax/cl , where vmax is the maximum velocity of motion of an oscillating source; and since ω1b/cl > 1, the oscillating source emits sound effectively in the case where its maximum motion velocity is larger
222
MOVING THERMORADIATION SOURCES OF SOUND
than the corresponding velocity of sound in a medium. Moreover, differing from the case of a motionless monochromatic thermoradiation source, sound fields of a source performing an oscillatory motion are concentrated near the motion plane and they are absent in the plane perpendicular to the motion direction. In conclusion let us consider the particular features of sound generation by a thermoradiation source in the process of motion of a radiation spot along a circle x0(t) = B sin Ωt, y0(t) = B cos Ωt. In this case
F∗l (ω ) =
F∗t =
∞
ω J m m B sin θ exp(imϕ ) F (ω + mΩ) , c l m = −∞
∑
∞
ω J m m B sin θ exp(imϕ ) F (ω + mΩ) , c t m = −∞
∑
F∗ R (ω ) =
(8.25)
∞
ω J m m B exp(imϕ ) F (ω + mΩ) . v R m = −∞
∑
In the case of a very long pulse of penetrating radiation τ >> 2π/Ω, an expression for the spectrum F(ω + mΩ) can be written down approximately in the form F (ω + mΩ) = 2πδ (ω + mΩ) .
(8.26)
Substituting this expression into expression (8.25) and then into expression (8.17), we obtain an expression for sound fields,
σ RR =
2 τ 2 / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m m a − × ( 1 ) ∑ 2 2 2τ 2 ) 2cε R (1 + ω mτ χ )(1 + ω m µ m =1
{
2 ω J m m B sin θ ω m (τ µ K (θ ) + τ χ M (θ )) cos[ω m (t − R / cl ) + mϕ ] + cl
(
)
}
2 ω m M (θ ) − ω m τ χ τ µ sin[ω m (t − R / cl ) + mϕ ] ,
RADIATION ACOUSTICS
σ RR =
223
2 2τ 2 n a / 4) (3 − 4 / n 2 )αa 2 I 0 ∞ exp(−ω m (−1) m × ∑ 2 2 2 2 2cε R (1 + ω mτ χ )(1 + ω m τν ) m =1
{
ω 2 τ −τ J m m B sin θ V2 (θ ) ω m ( χ ν ) cos[ω m (t − R / ct ) + mϕ ] + c t (8.27)
}
2 ω m (1 + ω m τ χ τν ) sin[ω m (t − R / ct ) + mϕ ] ,
2 ∗ v R + − uz = V i 1 2 cε ρ (2v R ) 3 / 2 (πR)1 / 2 cl2 (3 − 4 / n 2 )αa 2 I 0
∞
∑
m =1
( −1) m
1/ 2
2 2 2 ω m exp( −ω m ) a / 4v R 2 2 (1 + ω m τ χ )[1 + (ω m / µv R )(1 − v R2 / cl2 )]1 / 2
V1∗ ×
ω J 0 m B × v R
π π R R + mϕ . + mϕ + ω mτ χ cos + ω m t − sin + ω m t − 4 4 vR vR Naturally, proceeding from general physical concepts, the directivity patterns of a thermoradiation sound source performing motion along a circle at a certain frequency are the product of expressions describing the directivity patterns of a motionless monochromatic thermoradiation source and the directivity patterns of sound radiation by a body moving along a circle (see Section 2 of this chapter and [148]). In this case as in the previous ones, the amplitudes of sound fields at the frequency Ω at a certain circle radius B may be larger than the amplitudes of sound fields from a motionless monochromatic radiation-acoustic source. The results obtained above can be extended readily to the case in which a beam of penetrating radiation moves along a circle and radiation intensity is modulated monochromatically with sound frequency ω0. If we assume that the length of radiation pulse is large compared with both the modulation period τ >> 2π/ω0 and the period of beam rotation τ >> 2π/Ω, i.e., if we take the quasi-monochromatic mode of sound radiation, then
224
MOVING THERMORADIATION SOURCES OF SOUND
sound waves with frequencies of the form ωm = ω0 ± mΩ, where m = 0, 1, 2, …, are emitted, apart from the waves at the frequencies multiple to the rotation frequency mΩ with the characteristics described by expressions (8.27). Summarizing, we can indicate the following particular features of sound generation in a solid by a beam of penetrating radiation moving at its surface. In the case of uniform and rectilinear motion of the beam and pulsed operation mode of a source of penetrating radiation, the characteristics of generated sound fields are the same as in the case of a motionless beam but with a certain effective shape of radiation pulse. In the Cherenkov directions (each kind of sound field has its own Cherenkov direction), a sound pulse is generated with the shape of envelope independent of the shape of the pulse of penetrating radiation and dependent on the parameters of the region of effective heat release and the observation angle. The amplitude of this pulse is proportional to the energy of radiation pulse. In the case of a very long pulse, the amplitude of the acoustic oscillations generated in the Cherenkov direction increases up to the rise of shock waves. Thus, for example, the increase of the amplitude of Rayleigh wave is determined by the expression 2 (3 − 4 / n 2 )αa 2 I 0 t ∗ v R uz = V2 + i1 − 2 c R 8cε ρ (πv R ) 3 / 2 l ∞
ω exp( −ω 2 a 2 / 4v R2 )
∫ [1 + (ω / µv 0
1/ 2
2 2 1/ 2 ](1 + ω 2τ χ2 ) R )(1 − v R / cl )
V1∗ ×
×
(8.28)
π π R R − t dω . − t + ωτ χ cos − ω sin − ω 4 4 vR vR This effect was predicted theoretically by Dykhne and Rysev [92] and discovered experimentally by Velikhov, Dan’shchikov, and Dymshakov [48]. In the particular case of intra-pulse quasi-monochromatic modulation, its presence does not influence the characteristics of sound fields in the Cherenkov directions and the observation directions close to them. A quasimonochromatic component at the Doppler frequency arises in sound fields only at certain angular distances from the Cherenkov directions.
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In the case of oscillatory motion or rotation of a beam of penetrating radiation and a long enough radiation pulse, sound disturbances are generated at frequencies multiple to the frequency of oscillatory movements or rotation frequency, respectively.
CHAPTER 9
Sound Generation by Single High-Energy Particles Theoretical studies of sound generation by single high-energy particles have become important now in connection with possible practical applications (e.g., the DUMAND Project). The history of this problem has been considered already partly in the Introduction. This chapter treats sound generation by single elementary particles in various model situations, i.e., in an infinite space, in a solid half-space (bulk waves), at a solid surface (the Rayleigh wave), and efficiency of sound generation in an infinite space. We consider only the particles which give birth to cascade showers in the process of absorption in a substance, since single particles with energy insufficient for production of showers generate very weak acoustic fields. As geometrical dimensions of a cascade shower are greater than 10−5 m, we may ignore the effect of heat conductivity on sound generation under these conditions.
1. SOUND GENERATION BY A PARTICLE IN INFINITE SPACE Formulation of the problem of sound generation by single high-energy particles in infinite space (i.e., when the distance from the region of sound generation to the free boundaries of a body is large and we may ignore waves reflected from the boundaries) is possible for example in the case of 227
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SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES
high-energy neutrino and muons, which have large penetrating capability and can produce nuclear-electromagnetic cascades deep within a substance. Figure 9.1 gives a scheme of rise of the cascade in a substance in the process of absorption of a single high-energy particle. We take equations (6.9) as initial in this case, i.e., 1 ∂2 (3 − 4 / n 2 )α div F div u = , ∆ − ∆Qdt − ∫ 2 2 2 c ρ ε c ∂ t c ρ l l 2 ∆ − 1 ∂ rot u = − rot F , ct2 ∂t 2 cl2 ρ
where F is the dynamic force, F = Q/c; Q is the energy released per unit time within unit volume; and c is the velocity of light in vacuum. We consider a homogeneous and isotropic solid space bearing in mind the fact that the result for a liquid space may be obtained readily as a particular case.
Figure 9.1 Rise of a nuclear-electromagnetic cascade in a substance as the result of absorption of a high-energy particle.
We take the dependence of the function of energy evolution Q on time in the form of a delta-function since the time of energy evolution is much smaller than other characteristic times, and the spatial dependence of the function Q is approximated by the expression
Q ( x, y , z ) =
x2 + y2 exp − a2 πa 2
µE
exp(− µz ) Θ( z ) ,
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where E is the cascade energy, 1/µ is the effective cascade length, a is the effective cascade radius, and Θ(z) is the Heaviside function. The origin of coordinates is selected in this case in the point of cascade rise and the axis z is directed along the cascade axis in the direction of its development. Figure 9.2 dives an idea of the geometry of the problem. We may expect that in the process of problem solution, such approximation for the function of energy evolution Q(x, y, z) would provide an opportunity to obtain basic features of a sound field generated by a particle. This approximation is very rough because of the fact that it does not take into account the increase of the cascade radius in the process of its development.
Figure 9.2 Geometry of the problem of sound generation in an infinite space by a single high-energy particle.
First, let us consider the far sound field generated in an infinite medium by a nuclear-electromagnetic cascade. Using the technique for solution of boundary problems, which has been developed in Chapter 6, we can write down expressions for the components of a stress tensor in a sound field in a solid elastic medium,
σ RR = −
(3 − 4 / n 2 )αE 8πcε Rτ µ2
exp
R − cl t Erfc τ a − R − cl t , exp − 2τ µ cl τ µ cl τ a 4τ µ2
τ a2
(9.1)
σ Rθ =
R − ct t τ2 Erfc τ a − R − ct t , exp a exp − 2τ µ ct nτ a ct nτ µ 8πcRct n 2τ µ2 τ µ2 E sin θ
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SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES
where σRR and σRθ are the components of stress tensor, which originate from longitudinal and transverse waves, respectively; R and θ are the spherical coordinates of the distance from the cascade to the observation point and the angle between the cascade axis and the direction to the observation point; τa = a sinθ/cl and τµ = cosθ/µcl ; and other notations correspond to the ones introduced earlier. The near wave field of longitudinal waves in the case of observation in the direction perpendicular to the cascade axis is described by the following expression:
σ RR = −
c (3 − 4 / n 2 )α µE l 4πcε πR
2 2 ω a exp − × ω ∫ 2 4 c 0 l
∞
(9.2) π R cos − ω − t dω . 4 cl Parameters of a cascade produced by a high-energy particle are approximately equal for various liquid or solid media if the values of density and charges of elements constituting these media are close. Thus, according to the data by Askarijan et al. [194], the parameters of a cascade 15 from a neutrino with energy of the order of magnitude of 10 eV in water are 1/µ ≈ 4 m and a ≈ 2 cm. In this case the estimation of effective sound pressure (in Pa) in the near wave field (f ≈ 30 kHz) is determined according to expression (8.2) as peff ≈ 0.1
E 1 , E0 R
(9.3)
where and below R is the distance (in meters) and E0 = 1016 eV. Formula (9.3) corresponds to the analogous expression for estimation of the level of sound signal in the near wave field given by Berezinskii and Zatsepin [24]. If we take Antarctic ice for example as the medium, where a cascade arises in the process of absorption of a high-energy neutrino, the parameters of a cascade are approximately the same there as in water. In this case if we take the next numeric values: cl = 4⋅103 m/s, n = 2, α = 5⋅10−5 s−1, and cε = 2⋅103 J/(kg⋅C) that correspond to the ambient temperature −20°C [178], then the estimation of effective sound pressure in the near wave field (f ≈ 90 kHz) is given according to expression (9.2) by the next relationship:
RADIATION ACOUSTICS
eff σ RR ≈
231
E 1 . E0 R
(9.4)
According to formulae (9.3) and (9.4), the effective sound pressure in ice is approximately one order of magnitude higher than the effective sound pressure in water, other conditions being equal. This difference is caused by the fact that the Gruneisen parameter for ice is approximately one order of magnitude larger than that for water (Γ = αc2/Cp). In the case of the far wave field at τa >> τµ, i.e., in the case of observation almost perpendicularly to the cascade axis, we obtain the next expression for longitudinal waves:
σ RR = −
2 E (3 − 4 / n 2 )α 1 R − cl t ( R − cl t ) exp − cε cl2τ a2 2π π τ a2 R cl τ a
.
(9.5)
Correspondingly, the estimation of effective sound pressure (in Pa) in ice under the conditions considered above is eff σ RR ≈ 0.1
E 1 . E0 R
(9.6)
An analogous estimation for effective sound pressure in water is approximately one order of magnitude smaller as in the case of the near wave field. The boundary between the far and near wave fields for the parameters of the cascade given above and the effective frequency f = 30 kHz for water and f = 90 kHz for ice lies at the distance R ≈ 100 m. In this case the level of sound pressure in ice at a distance 100 m in the observation direction perpendicular to the cascade axis is 10−3 Pa approximately, and the level of sound pressure in water is 10−4 Pa for a particle with the energy of the order of magnitude of 1016 eV. The far wave field at τa c t . l c |τ | |τ | cl | τ µ | µ l µ Effective sound pressure in ice for the cascade parameters given above and observation directions close to the cascade axis is (f ≈ 1 kHz) eff σ RR ≈ 10 − 5
E 1 . E0 R
(9.8)
Correspondingly, sound pressure in water under analogous conditions is approximately one order of magnitude smaller than effective sound pressure in ice. It is interesting to note that the level of effective sound pressure in the far wave field in the observation direction perpendicular to the cascade axis (θ = 90°) is approximately four orders of magnitude higher than the level of effective sound pressure in the case of observation along the cascade axis (θ = 0°). The same may be said with respect to sound pressure in water. As for the level of shear stress caused by transverse waves and originating from the dynamic mechanism of sound generation, according to expression (9.1) it is approximately five orders of magnitude smaller than the corresponding level of pressure caused by longitudinal waves for the same observation angles. And moreover, the angular factor sinθ is present in the expression for shear stress in the far wave field, which exists due to transverse waves. It is very interesting in this case that in the far wave field, the level of sound pressure produced by longitudinal waves at small θ is of approximately of the same order of magnitude as the level of shear stress produced by transverse waves at θ = 90°. A schematic shape of a sound pulse in the far wave field, which is produced by longitudinal waves and described by expression (9.5), is given in Fig. 9.3a. Analogously, the shape of a sound pulse in the near wave field, which is described by expression (9.2), is given in Fig. 9.3b. Figure 9.4 presents the maximum values of the relative level of the component of stress tensor σRR as a function of the observation angle θ according to general expression (9.1). As for the component of stress tensor σRθ , according to expression (9.1) its relative levels differ from the relative
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233
levels of the component of stress tensor σRR only in the additional factor sinθ.
Figure 9.3 Shapes of sound pulses produced by longitudinal waves in the (a) far and (b) near wave fields in the case of observation in the direction perpendicular to the cascade axis.
Figure 9.4 Maximum values of the component σRR as a function of observation angle.
Berezinskii and Zatsepin [24], Askarijan et al. [194], and others considered the near wave field with cylindrical symmetry, which arises in an infinite liquid space from a nuclear-electromagnetic cascade. It has been noted that using a set of receivers of sound pressure, it is possible to determine the direction of the cascade axis in space but it is impossible to say where a particle producing this cascade has come from, from “above” or from “below”. Observing the far wave field under different angles, it is possible to determine the direction where a particle has come from. Naturally, in the case of a single receiver, the situation when a particle comes from “above” and the receiver is positioned “below” a cascade, and the situation when a particle comes from “below”, while the receiver is positioned “above” a cascade, are indistinguishable in principle. However, in the case of two receivers such situations become distinguishable. The presence of transverse waves in a solid apart from longitudinal waves can provide additional information on the distances from a cascade to the reception point. However, the level of transverse waves produced due to
234
SOUND GENERATION BY SINGLE HIGH-ENERGY PARTICLES
the dynamic mechanism of sound generation is much smaller than the corresponding level of longitudinal waves. But if it is possible to detect them, this is an additional information channel. We have to keep in mind that various inhomogeneities may exist in real solids. Scattering of longitudinal waves into transverse ones at these inhomogeneities masks the dynamic mechanism of sound generation.
2. SOUND EXCITATION BY A SINGLE PARTICLE IN A SOLID HALF-SPACE Let a high-energy particle be incident upon the surface of a solid half-space. It is assumed that a cascade arises directly at the half-space boundaries. The function of energy release Q in this case is expressed in the following way:
Q ( x, y , z , t ) =
y2 + x2 exp − a2 πa 2
µE
exp(− µz )δ (t ) ,
(9.9)
where E is the energy of the cascade or the initial particle (under the condition that the total energy of the initial particle is transferred to the cascade and not taken away by some particles which do not take part in the rise of the cascade, e.g., muons). Analogously to the way it has been done in the case of sound generation by short pulses of penetrating radiation, we can write down expressions for the components of stress tensor in the far wave field,
σ RR =
(3 − 4 / n 2 )αE 8πcε Rτ µ2
2 τ exp a 4τ µ2
R − cl t V1 (θ ) exp c τ × l µ
τ R − cl t R − cl t Erfc τ a − R − cl t , − exp − Erfc a + 2τ µ clτ µ 2τ µ clτ a clτ a
σ Rθ =
(3 − 4 / n 2 )αE 8πcε Rτν2
n 2τ 2 a V2 (θ ) exp 4τ 2 ν
exp R − ct t × cτ t ν
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235
nτ R − ct t , × Erfc a + 2 ct nτ a τ ν (9.10)
σ Rϕ =
2 τ exp a 4τ µ2 8πcRct n 2τ µ2 E sin ϕ
τa R − ct t R − ct t exp − c nτ Erfc 2τ − c nτ − t a µ t µ
R − ct t Erfc τ a + R − ct t . exp ct nτ µ 2τ µ ct nτ a As it has been done in the case of sound generation by short pulses of penetrating radiation, we can write down expressions for the components of stress tensor in the case when τa >> τµ and nτa >> |τν|,
σ RR = −
2 (3 − 4 / n 2 )αE R − cl t ( R − cl t ) exp − 2π π cε Rτ a2 cl τ a cl2τ a2
M (θ ) +
2( R − c t ) 2 τ 2 µ exp − ( R − cl t ) K (θ ) , l − 1 τ 2 2 cl2τ a2 cl τ a a (9.11) (R − c t) 2 (3 − 4 / n 2 )αE R − ct t t σ Rθ = − exp − 2 2 2 2 2 c n τ 2π π cε Rn τ a t a ct n τ a
σ Rϕ =
V (θ ) , 2
(R − c t)2 R − ct t t exp − c 2 n 2τ 2 π π cRct n 2τ a2 ct nτ a t a E sin ϕ
,
and in the case when τa |τν|.
Figure 9.6 Shape of dependent pulses in the case of observation under small angles: (a, b) τa < τµ and (c) nτa < |τν|; (1) θ < arcsin (1/n) and (2) θ ≥ arcsin (1/n).
Shapes of sound pulses for the components of stress tensor σRR, σRθ, and σRϕ in the case τa >> τµ and nτa >> |τν|, i.e., the case described by expressions (9.11), are shown in Fig. 9.5. Figure 9.6 presents shapes of sound pulses for the components σRR and σRϕ in the case τa 6 − 4⋅21/2 ≈ 0.35. If the region of heat evolution satisfies the conditions (µ/k)2 > 1, where τ is the length of laser pulse, which is determined by the function f(t); QR(R′) = µAI(x − vt, y) exp(−µz). We restrict ourselves to consideration of the particular features of the acoustic field of a laser thermooptical sound source in the far wave zone. In other words, as in Chapter 8 we consider the field at the distances R >> L, where L is the length of the trajectory of laser beam motion. According to previous exposition (see also [31, 112, 147, 148, 240]) we can write down the next space-time representation for sound signals from a moving laser thermooptical sound source, ∞
p0 (R, t ) = −
iωα ∗ (k ) f (ω − β ∗ω ) exp − iω t − R dω , QR 4πC p R c −∞
∫
∞
p D (R, t ) = −
∫
−∞
(10.6)
iωαm ∗ Q R (k ) f ∗ (ω − β ∗ω ) × 4πC p R (10.7)
R exp − iω t − dω , c where p0(R, t) is the pulsed contribution and pD(R, t) is the Doppler contribution into the field of a moving laser thermooptical sound source, k=
R ωu ∗ (k ) = Q ( R )e ikR dR , f (ω ) = 1 , u = , QR ∫ R c R 2π
∫ f (t )e
i ωt
dt .
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259
The length of the sound train of the Doppler-compressed envelope is τD = τ |1 − β*|. The characteristic scales due to spatial distribution of a source are τa = c−1a sinθ and τµ = (µc)−1cosθ. Depending on a relation between the time τD determining the spectral width of the functions f(ω(1 − β*)) and the times τa and τµ determining the spectral width of the function QR*, the shape of sound signal is determined either by the Doppler-compressed envelope or by the spatial distribution of the source [147]. First, we assume that τD >> τa, τµ, 2πΩ. Then we can consider the function QR*(k) in the expression for the Doppler signal (10.7) changing smoothly as compared to the function ωf and take it outside the integral at the value ω = Ω(1 − β*)−1. In the case of a supersonic source, if the observation point is located within the angular interval 0 ≤ θ < θm, the major contribution into expression (10.7) belongs to the integral with respect to the negative ω. In this case the projection of the source velocity onto the observation direction is larger than the sound velocity (β* > 1). Consequently, the sound signal arrives at the observation point in the reverse sequence as compared to the sequence of its generation by the source. Therefore, this Doppler signal may be called anomalous. The modulus sign should be substituted for it in the expression for the Doppler frequency. In the case of subsonic motion of a source or in the interval θm < θ ≤ π in the case of supersonic motion, the anomalous Doppler signal paD is absent and there is only the normal Doppler signal pnD. The Doppler signals are described by an expression [33] p(n,a)D ( R , t ) =
mAα |J 4πC p R
R R | Jf ′ J t − − iΩ D f J t − − c c (10.8)
∗ ( k ) exp[i ( k R QR D D − Ω D , t )] , ∗
where J = (1 − β )−1 is the Doppler factor, kD = uΩD/c, and f ′(ξ) = df/dξ. The contribution proportional to f′(J(t − R/c)) is caused by tuning of moving sources connected with the fact that the envelope changes with time. This contribution is small as against the second one. In the case of a rectangular envelope in the intermediate time moments f ′(t) ≡ 0, it arises only at the end points of the source trajectory. Formula (10.8) for a rectangular envelope was obtained by Lyamshev and Sedov [148] and Bozhkov, Bunkin, and Kolomenskii [31]. It follows from expression (10.8) that the amplitude of the Doppler signal produced by a moving source can be obtained from a corresponding expression for a motionless source. In this case it is enough to substitute the
260 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION
components of the Doppler wave vector uΩD/c for the components of the wave vector uΩ/c and multiply the expression obtained by the modulus of the Doppler factor. This agrees with a general rule, which allows us to describe the sound field of a moving volume-distributed source in the far wave field proceeding from the equivalent motionless source [240], which is obtained from the initial one by compression of the time scale t → Jt and multiplication by the modulus of the Doppler factor. Therefore, sound excitation by a scanning laser beam occurs in the same way as by a motionless source but with the time-compressed envelope |J|f(Jt) and modulation frequency equal to the Doppler frequency. This rule as applied to pulsed excitation of sound by a scanning laser beam was formulated by Lyamshev and Sedov [148]. Using this rule, it would be possible to obtain immediately formulae analogous to expression (10.8) proceeding from calculations of sound excitation by a motionless laser thermooptical sound source, as has been stressed above (Chapter 8, Section 1). Let us consider the most typical cases, which are realized at different relations between the Doppler length of a laser pulse τD and times of sound travel along the projections of longitudinal and transverse dimensions of a laser source onto the observation directions τµ and τa. Let τD >> τa, τµ that corresponds to excitation of the sound pulses long in comparison with the times τa and τµ. This case is realized at the velocity of source motion satisfying the conditions |1 − β*| >> τa/τ, τµ/τ almost in the whole half-space z ≥ 0 if long laser pulses are used, and in the region of manifestation of the anomalous Doppler effect if one uses short pulses. The inverse Fourier transformation applied to formula (10.6) gives an expression for pressure in an excited sound pulse under given approximation accurate to the terms of the order of magnitude of O(τµ3/τD3), p0 ≈
Aαa 2 I 0 R τ µ | J |3 f ′′ J t − , c 2C p R
(10.9)
i.e., in the case of large values of Doppler length under these conditions, the shape of sound signals in the far wave field is determined by the second derivative of the effective time envelope of a laser pulse, and pressure is proportional to the light intensity and inversely proportional to the absorption coefficient of laser radiation. At τµ >> τD >> τa that may be realized in the case of the “rod-like” shape of a source, the projection of the motion velocity of a laser thermooptical sound source onto the observation direction vR satisfies the condition τµ/τ >> |1 − vR/c| >> τa/τ (this may require supersonic motion of
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261
the source at the surface of a liquid). Pressure in a sound pulse is determined by the expression
p≈−
| t ′ | sf Aαa 2 I 0 . | J | f ( Jt ′) − exp − τ µ 2C p Rτ µ 2τ µ
(10.10)
The shape of a sound pulse is determined by the envelope itself in this case:
| J | f ( Jt ′) , t ′ = t −
R , sf = c
∞
∫ f (t )dt .
−∞
Let us consider the case τD τa, τµ at small subsonic velocities of a source was realized in the whole half-space occupied by the liquid, while at velocities close to the sound velocity it was realized beyond the region of the observation directions close to the line of motion, i.e., at vR/R > a, µ−1, and the observation point is located in the far wave field with respect to the trajectory length and the source dimensions a and µ−1. The
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263
shape of the sound pulse corresponds to the second time derivative of the envelope of the laser pulse given in Fig. 10.17a (the time scale is compressed 34 times approximately). The total length of the compression pulse in the sound signal ∆τ = 16 ± 2 µs is determined by the time interval between the points of inflection at the fronts |J|f(Jt), which coincides to a good precision in the case under consideration with the length of the “effective” laser pulse at the half-height τD = 14.4 µs. Such coincidence of experimental and calculated data was obtained for the whole considered range of values of τD. The amplitude of the rarefaction phase of the sound signal given in Fig. 10.16 was 45 ± 10 Pa and that of the compression phase was 30 ± 10 Pa.
Figure 10.16 Oscillogram of a sound pulse excited at τD >> τa, τµ [33].
Figure 10.17 Shape of envelope of laser pulse (a) without modulation and (b) with harmonic modulation of light intensity at the frequency 100 kHz [33].
If the observation direction was changed, the length of a sound signal changed correspondingly to the change of the Doppler factor J = (1 − β* cosθ ′)−1. Namely, when θ ′ increased from 3 to 8.5°, ∆τ increased from 6 to 16 µs, while the calculated value of τD increased from 7.6 to 14.4 µs and τµ increased from 2.1 to 6 µs at v = 0.985 c. The amplitude of sound pressure, which is determined as |J|3f′′(Jt) according to expression (10.9), changes proportionally to |J|3 ~ ∆τ−3.
264 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION
Moreover, sound pressure within the considered range of τD is proportional to τµ, which grows with increase of θ. Figure 10.18 gives the dependence of the amplitude of sound pulses on the parameter τµ/∆τ3 in the case of changing of the observation angle from 26.2 to 19.3° and the constant value ϕ = 17.7°. In this case τµ changed from 14.4 to 6 µs and the measured length of rarefaction pulses changed from 26 to 3.4 µs. A linear dependence of sound pressure on the parameter τµ/∆τ3 was observed in a broad range of angles θ ′ and ϕ, for which τD >> τa, τµ. A certain deviation of experimental data from the theoretical linear dependence at small θ ′ (see Fig. 10.18) is connected with the fact that in this case τD becomes comparable to τµ and the condition τD >> τa, τµ gets violated.
Figure 10.18 Dependence of amplitude of sound pulses on the parameter τµ/∆τ3 in the case of change of observation direction [33].
The value of τD becomes comparable to the values of τµ and τa for the observation directions satisfying the condition |1 − β*| ~ τa/τ, τµ/τ. It is necessary to note that this condition is realized for any selected observation direction R if vR is close enough to c that may require supersonic motion of a source at the liquid surface. Measurements demonstrated that the shape of sound pulses in this case differed from the shape of pulses considered earlier for the case τD >> τa, τµ. The amplitude and length of sound signals under these conditions depend on the source velocity to a less extent. The cubical dependence of the amplitude on J transforms into a linear one. If the motion velocity of a laser thermooptical sound source increases further and the value of τD decreases, a transition to the case τµ >> τD >> τa occurs. This case is realized at τµ/τ >> |1 − β*| >> τa/τ. Sound pulses excited under these conditions are described by approximate expression (10.10). A rod-like source configuration used in experiments provided an
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265
opportunity to fulfill such observation conditions when τµ ≈ 2τD ≈ 4τa at v = 1.003 c close to ϕ = 0 and θ ′ = 13.3°. An oscillogram of a pulse detected under such conditions is given in Fig. 10.19a. The pressure trend in the rarefaction pulse repeats the shape of the effective envelope |J|f(Jt) as formula (10.10) predicts. The pressure amplitude in the rarefaction pulse is 102 ± 30 Pa. The amplitude of positive splashes described by the second term in formula (10.10), which is proportional to the small quantity τD/τµ, is equal to 20 ± 10 Pa in this case. The length of the rarefaction pulse at its half-height (the pulse is given in Fig. 10.19b) is ∆τ = 5.3 ± 2 µs that corresponds to the calculated value τD = 4.7 µs. Further reduction of the Doppler length leads to the fact that τD becomes much smaller than τa and τµ, i.e., τD τa and (b) τD < τa, τµ [33].
Now let us consider the particular features of sound fields in the case of uniform motion of a source with velocities equal to or exceeding the sound velocity in a medium. If v > c, three characteristic cases are realized depending on the position of the observation point: vR < c, vR = c, and vR > c. All results given above for the case of radiator motion with subsonic velocity stay the same in the region of observation directions with vR < c for sound fields of a radiator moving with supersonic velocity.
266 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION
An important feature of radiator motion with supersonic velocity is the presence of observation directions where vR = c. In-phase accumulation of sound disturbances from all parts of the trajectory of radiator motion occurs in these directions in the far wave field. Due to this fact it is possible to increase energy release and form especially intense and short acoustic signals, which are called the Mach wave in acoustics, increasing the length of the light track L. Within the framework of this description, the length of the light track L is restricted only by the rise of nonlinear acoustic phenomena at large values of L and high intensity of sound. Formation of the Mach wave is demonstrated by the plots given in Fig. 10.20. These plots characterize the dependence of the amplitude and length of sound pulses on vR. They are obtained for a fixed position of observation point R = 4 m, θ ′ = 6°, and ϕ = 0 in the case of the velocity of radiator motion changing in the range 0.9 – 1.15 c. As the value of vR approaches c, 3 the amplitude of sound grows proportionally to |J| within the region τD >> τa, τµ, then it slows down to linear dependence on |J|, and at τD > τa, τµ and to a value about max(τa,τµ) at τD c. The difference in the amplitude and length of the pulses arises because of the presence of the directivity properties of a rod-like source. Under the conditions corresponding to Fig. 10.23 τD > R. The condition λR ~ a was satisfied in the experiments, where λR is the Rayleigh wavelength. The first condition meant that the experimental conditions
RADIATION ACOUSTICS
279
satisfied the requirements of one-dimensionality of the problem of SAW excitation. This simplified comparison of experimental results and estimations. In the case when the second condition was satisfied, the efficiency of SAW excitation was close to optimal. Light exposure was performed by pulses of radiation of a YAG-Nd3+ laser at the fundamental wavelength (λ1 = 1.06 µm and hν1 = 1.17 eV) and the second harmonic wavelength (λ2 = 0.53 µm and hν2 ≈ 2.34 eV). The pulse length was τL ≈ 20 ns. The main experimental result was the change of polarity of detected SAW in the process of changing the wavelength of radiation absorbed in silicon. It is presented in Fig. 10.33. The analysis shows that this effect is connected with the change of the dominant mechanism of SAW excitation.
Figure 10.33 Oscillograms of detected acoustic pulses for different wavelengths of incident optical radiation [1]. (a) λ1 = 1.06 µm, scale 0.2 µs/div.; (b) λ2 = 0.53 µm, scale 0.5 µs/div.
In conclusion let us give some results of measurements of directivity patterns of laser thermooptical sound sources in solids. These results were published in several papers (see [186] and especially [222]). In experiments [222] acoustic signals were excited in aluminium semispherical samples. The experimental scheme is shown in Fig. 10.34. A beam of a pulsed YAG-Nd3+-laser operating in a Q-switched mode at the wavelength 1.06 µm was incident perpendicularly on the plane surface of a solid aluminum semi-sphere. The pulse length was ~50 ns and the pulse energy was ~30 mJ. Measurements of longitudinal and shear waves (the radial and tangential displacements of the semi-sphere surface, respectively) were performed with the help of piezoelectric sensors of corresponding polarization. Piezoelectric sensors were equipped with a special clamping device and a thin oil film was deposited at the sensitive surface of a sensor contacting the semi-spherical surface of a sample in order to improve
280 EXPERIMENTAL STUDY OF THERMORADIATION EXCITATION
acoustic contact. A high-quality acoustic resonator tuned to a fixed frequency 1 MHz was installed at the sensor output.
Figure 10.34 Scheme of setup for measurement of directivity patterns of laser thermooptical sound sources in solids. (1) An aluminium (solid) semi-sphere with diameter 10 cm; (2) piezoelectric sensor; (3) laser beam; (4) laser beam splitter; (5) Q-switched laser; (6) photodiode; (7) oscilloscope input [222].
Figure 10.35 Experimentally measured directivity patterns of a “linear” laser thermooptical sound source in an aluminium sample [222]. (a) Longitudinal vibrations and (b) transverse vibrations. Points indicate experimental data.
A laser beam was focused by lenses at the sample surface either into a spot with diameter from 1 to 3 mm or into a line with width 0.025 cm and length 8 cm. The condition of the far wave field of a laser thermooptical sound source was satisfied for the diameter of aluminium semi-sphere. Figure 10.35 shows the directivity characteristics of a “linear” laser thermooptical sound source with the dimensions indicated above, which were obtained in the result of measurements at the frequency 1 MHz for longitudinal and transverse waves. Analogous characteristics obtained on
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the basis of calculations are given in Fig. 10.36. One can see that theoretical and experimental results agree quite well at least qualitatively.
Figure 10.36 Theoretical dependences characterizing directivity of a “linear” laser thermooptical sound source in an aluminum sample for (a) longitudinal and (b) shear waves [222].
6. SOUND EXCITATION BY X-RAYS (SYNCHROTRON RADIATION) IN METALS The results of the first experimental studies of sound excitation by X-rays in condensed matter and first of all in metals, were presented by Kim and Sachse [226, 227]. These results are given in the most complete form in the paper by Sachse, Kim, and Pierce [250]. There is nothing surprising in utilization of X-rays for sound excitation in condensed media and especially in metallic targets after studies of laser excitation of sound. In both cases electromagnetic radiation is used to generate sound. However, in contrast to laser radiation, the track length of quanta of X-rays in metallic targets depends on physical parameters of a target (the atomic number of a substance) and the energy of radiation quanta. For example, the penetration depth of X-rays with energy 10 keV in aluminium is about 1.4⋅10−2 cm while it is about 3.76⋅10−7 cm in the case of laser radiation. This difference may be essential from the point of view of applications. The source of X-rays in the experiments [226, 227, 250] was the synchrotron of Cornell University (USA). Synchrotron radiation of X-ray range was used. We should recall that synchrotron radiation arises when particles such as electrons and positrons for example are accelerated in a vacuum chamber of an accelerator up to relativistic velocities in the process
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of their motion along a curved trajectory in a magnetic field. Synchrotron radiation used in the experiment [250] was generated in the process of motion of a “package” of electrons in the storage ring of the synchrotron. Fundamental characteristics of synchrotron radiation are high intensity, broad spectral range, strong polarization, pulsed time structure, and natural collimation. Generated X-rays become homogeneous and wide-band at the energy of about 10 keV. As the energy of accelerated particles grew, these properties of radiation were violated. X-rays used in experiments had the following parameters: the pulse length 0.16⋅10−9 s and the pulse repetition period about 2.56⋅10−6 s. This pulse periodicity was determined by the time of motion of an electron package in the storage ring of the synchrotron. The energy of the beam of photons (per pulse) of X-rays incident directly at a sample-target was 1.12⋅10−6 J per pulse. Figure 10.37 presents the block scheme of the experiment. X-rays (synchrotron radiation) were directed through (1) a collimator at (2) a protecting screen and then at (3) a sample-target. (4) A wide-band piezoelectric transducer was fixed at the opposite side of the target. A signal from the transducer output was fed to the input of (5) a preamplifier with the band 0.01 – 2 MHz and then to (6) an integrating amplifier and (7) data collection system. A synchronization signal was used to improve noise stability of measurements. This signal was taken from (8) an X-ray detector, which had a response time not larger than 10−9 s. The synchronization signal was fed to (6) the integrating amplifier containing an analog-to-digital converter. The minimum dimension of the X-ray beam at the target surface was 3 mm.
Figure 10.37 Block scheme of experimental set-up [250].
Sample-targets had the shape of a disk with diameter 5.72 cm and thickness 1.52 cm. They were made of aluminum, stainless steel, copper, bronze, and titanium. Aluminum cylindrical blocks with length 10.2 mm were used to measure the directivity characteristics of thermoradiation sound source in solids. It is expedient to note that the thickness of the disk
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targets was essentially larger than the radiation length, i.e., the penetration depth of X-rays in the target material (as it was mentioned above, in the case of aluminum µ−1 ≈ 1.4⋅10 cm). The characteristic times of sound propagation along the dimensions of a thermoradiation sound source in targets were larger than the length of X-ray pulses. In other words, the conditions of sound excitation in solids by very short pulses of penetrating radiation were satisfied. Figure 10.38 presents a typical recorded shape of a signal obtained from a wide-band transducer after digital filtration. Sampling time started at the arrival of a pulse of X-rays at the surface of a sample-target. The letter p indicates an ultrasonic radiation-acoustic pulse caused by longitudinal waves, and the letter s indicates the pulse produced by transverse waves; s(−1) corresponds to a pulse of transverse waves caused by the action of a preceding pulse of X-rays incident on a target and p(1) corresponds to a pulse of longitudinal waves cased by the next pulse of X-rays. We should remember that the repetition period of X-ray pulses was 2.56 µs. Check measurements of ultrasonic signals excited in a sample-target were conducted with the help of a piezoelectric transducer fixed at the lateral surface of a target and not at its rear side in the target center on the axis of X-ray beam incident at the target. This was done in order to determine once more that ultrasonic pulses were excited in the target and not in the piezoelectric detector located at the path of the X-ray beam.
Figure 10.38 Recording of signals from a thermoradiation sound source in a stainless steel target at the output of a piezoelectric transducer after wide-band digital filtering and derivation: p – pulses of longitudinal waves, s – pulses of transverse waves [250].
The dependence of amplitude of an ultrasonic pulse on energy in a beam of X-rays incident at a steel target is shown in Fig. 10.39. The vertical axis presents the values of voltage proportional to the amplitude of ultrasonic signal and the directions characterizing energy in the beam of X-rays are
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plotted in the abscissa. The beam energy is directly proportional to the amplitude of current in the particle beam in the synchrotron and electrical voltage is directly proportional to current in the synchrotron beam. The data were obtained as the result of averaging of at least one hundred single measurements (for each point). The analogous dependence of ultrasonic signal amplitude on the energy of photons of X-ray beam is given also in Fig. 10.40. The gap in the data in the area of the electrical voltage 0.7 V is caused by synchrotron operation.
Figure 10.39 Dependence of acoustic signal amplitude (longitudinal waves) generated by a beam of X-rays on the energy of photons in the beam [250].
Figure 10.40. Dependence of amplitude of acoustic signal (longitudinal waves) on energy in a beam of X-rays (notations are the same as in Fig. 10.39; another sampletarget) [250].
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Measurements of the dependence of amplitude values of an ultrasonic signal on the size of the aperture of an X-ray beam were conducted. Experiments were conducted in such a way that the beam was passing in front of a sample-target through a narrow slot. The slot height was constant and constituted 3 mm, while its length could be changed within the range from 3 to 20 mm. The results of measurements for a stainless steel target are given in Fig. 10.41. One can see that at the beginning a proportional dependence of peak value of amplitude of the ultrasonic thermoradiation signal on the slot length (aperture) was observed, and then this dependence was lost. Sachse, Kim, and Pierce [250] determined experimentally that in the case of significant increase of the slot length (the aperture of X-raybeam) the shape of ultrasonic signals (which stays almost constant in the process of initial change of the slot length) is subjected to considerable changes. The change of the shape of acoustic signals explains the loss of direct proportionality between the peak amplitude of ultrasonic signals and the linear dimension of the aperture of X-ray beam in the case of significant change of the aperture (slot) length.
Figure 10.41 Dependence of the maximum value of the amplitude of an acoustic signal of longitudinal waves in a sample-target on the dimensions of the aperture of an X-ray beam.
All presented experimental results agree quite well with theoretical conclusions. Indeed, it follows from theory that in the case of excitation of acoustic signals in a solid elastic half-space by very short pulses of penetrating radiation the amplitude of acoustic signal changes directly proportionally to the energy of a pulse of penetrating radiation. Just these conditions were realized in the experiment. The depth of penetration of Xrays into the target material was essentially smaller than the target
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dimensions and the radiation pulse length was very small as against the time dimensions of the thermoradiation sound source created in the target by the action of pulses of X-rays. It follows from theory also that, in the case of constant radiation intensity in a beam incident upon a target, the increase of the linear dimensions of the beam (slot) aperture must lead to the proportional change of the acoustic signal amplitude. Just such a dependence was observed in the experiments [250]. Sachse, Kim, and Pierce [250] measured also the directivity characteristics (angular dependence of acoustic signal amplitude) of a thermoradiation sound source in a solid. The results are given in Fig. 10.42. In the case corresponding to Fig. 10.42a, the beam dimensions were 3 × 3 mm2, and in the case given in Fig. 10.42b the dimensions were 3 × 15 mm2. These characteristics relate to longitudinal waves in a sample-target. As was noted by Sachse, Kim, and Pierce [250], the polar characteristics are very similar to the angular acoustic characteristics of an optoacoustic source arising in a solid under the effect of short pulses of focused laser radiation [222] (see Figs. 10.34 – 10.36).
Figure 10.42 Polar characteristics of a thermoradiation acoustic source in aluminium at different dimensions of a penetrating radiation beam. Measurements were conducted for longitudinal waves at the repetition frequency of pulses of X-rays 1.2 MHz [250].
7. SOUND EXCITATION BY A PROTON BEAM Experimental studies of sound excitation by a beam of protons in a condensed medium were conducted many times (see, for example, [84, 220, 253 – 255]. Here we will discuss in detail the results published by Danil’chenko et al. [84], Hunter and Jones [220], and Sulac [253].
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Sulac [253] conducted extensive experiments on sound excitation by proton beams in liquids at the accelerators of the Brookhaven National Laboratory (USA) and Harvard University (USA). The proton beam came from an accelerator. It was directed through a collimator into a water basin. The acoustic signal was detected by a hydrophone equipped with a preamplifier and a circuit for amplification and detection. Measurements were conducted in the pulsed mode. Experiments on sound excitation in water by a beam of protons with energy 200 MeV were conducted using the linear accelerator of the Brookhaven National Laboratory. The basin dimensions were considerably larger than the proton track length and spatial dimensions of an acoustical signal in water. This provided an opportunity to perform space-time selection of the direct acoustic signal and signals reflected from the basin walls. The proton track length in water was about 30 cm. The beam reset (pulse length) changed within the limits from 3 to 200 µs and energy release in a pulse varied within the limits from 1010 to 1021 eV. The diameter of the proton beam was constant and equal to 4.5 cm. Analogous experiments were conducted at the cyclotron of Harvard University with a beam of protons with energy 158 MeV. In this case the energy release in a pulse was 1015 eV, the pulse length was 50 µs, and the track length in water was about 16 cm. Cyclotron experiments were conducted not only with water but also with various liquids in different conditions (at different values of liquid temperature and static pressure). The dimensions of the basin in this case were smaller than those of the basin used in the experiments with the linear accelerator but still considerably larger than the characteristic sound wavelength. The third series of experiments was performed using the accelerator of the Brookhaven National Laboratory with a beam of protons with energy 28 GeV (and very small pulse length). Energy release in a pulse was smaller than 1019 eV as in the experiments with the linear accelerator. In a typical experiment of this series, 3⋅1011 protons covered the distance of 20 cm during a pulse, the beam diameter was varied from 5 to 20 mm, and the pulse length was smaller than 2 µs. Opposite to the cyclotron measurements, the length of a pulse of protons was always smaller than the time of sound propagation along the beam diameter, i.e., the conditions of sound excitation by a very short pulse of penetrating radiation were satisfied. The scheme of the experiments is shown in Fig. 10.43. The measurements demonstrated naturally that in the near wave field of a thermoacoustic array, an acoustic signal has the shape of an N-wave. The dependence of the N-wave period on the beam diameter was measured in the experiments with a beam of protons with energy 28 GeV (Fig. 10.44). One can see that the observed dependence is linear. This
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corresponds to the theoretical conclusion that under the condition τ < τa the length of acoustic signal is directly proportional to the beam diameter.
Figure 10.43 Scheme of sound generation by a proton beam. L is the track length of protons, d is the diameter of the region of a proton beam, R is the distance to the observation point, and A is the characteristic distance to the boundary between the near and far wave fields [253].
Figure 10.44 Dependence of acoustic signal length on the diameter of proton beam [253].
Figure 10.45 demonstrates the experimental data characterizing the dependence of acoustic signal amplitude on energy release in a pulse in the case of a proton beam of a very small duration (τ ≤ 10 µs). Theory predicts that the linear dependence of signal amplitude on the energy of penetrating radiation must be observed. This fact was confirmed in the experiments. Analogous data are given in Fig. 10.46 but for a proton beam with smaller energy release in a pulse. The experiments were conducted at the cyclotron of the Harvard University. It follows from theory that in the case of constant energy (power) in a beam the amplitude of acoustic signals must change in inverse proportion to the square of the beam diameter. The experimental data confirming this rule are given in Fig. 10.47. According to the equation of sound generation by penetrating radiation in condensed media, the amplitude of sound signals increases in direct
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proportion to the ratio of the coefficient of thermal volumetric (linear) expansion to the specific heat conductivity of a medium. Sulac confirmed this connection experimentally (see Fig. 10.48) [253].
Figure 10.45 Dependence of acoustic signal amplitude on energy release in a proton beam (H2O, 20°C, d = 4.5 cm, R = 100 cm) [253].
Figure 10.46 Dependence of acoustic signal on energy release in a proton beam (H2O, 20°C, d = 1 cm, R = 8 cm) [253].
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More evidence of the thermoradiation mechanism of sound generation by penetrating radiation under a moderate density of energy released in a medium is the dependence obtained of the amplitude of an acoustic signal of a thermoradiation sound source on temperature [253]. This dependence for a proton beam in water is given in Figs. 10.49 and 10.50. A particular feature of this experimental dependence is the fact that the coefficient of thermal expansion for water must turn into zero at 4°C. Meanwhile, the amplitude of an acoustic signal generated by a proton beam vanishes at approximately 6°C. This fact may be explained by the existence of an additional mechanism of sound generation, namely microstriction compression of a medium under the action of ionizing particles [10, 66]. This leads to the effect of compensation of thermal expansion of a medium caused by the thermoradiation mechanism at the temperature about 6°C [220].
Figure 10.47 Dependence of acoustic signal amplitude on the diameter of a proton beam [253].
Hunter and Jones [220] conducted thorough experiments in order to determine the presence of a non-thermoradiation mechanism of sound generation in water affected by a proton beam. Experiments were conducted at the accelerator of the Brookhaven National Laboratory with a beam of high-energy protons 20 GeV. The pulse length was varied in the range 1.5 ÷ 3 µs that is essentially less than the time of sound travelling along the beam diameter (8 ÷ 10 mm and 5 ÷ 7 µs, respectively). A proton beam was introduced into a special Dewar flask with water. Water temperature could
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be varied, and these changes were carefully monitored. Sound pulses were detected by a miniature high-sensitivity hydrophone made of the piezoelectric ceramics of zirconate-titanate, with the transmission band from 0.1 Hz to 120 kHz, and equipped with a circuit of amplification and detection.
Figure 10.48 Dependence of acoustic signal on the ratio α/Cp [253].
The purpose of the experiments was to determine how the shape of a sound pulse changes in the process of changing of water temperature within the temperature interval close to the critical temperature where the coefficient of thermal expansion of water turns into zero and then changes its sign for the opposite one. A preliminary judgment that the power spectra of signals of thermoradiation and non-thermoradiation “origin” under certain conditions (in their major energy-carrying part) must differ since the shape and length of pulses of different origin is different, was made on the grounds of the idea on a possible non-thermoradiation (non-thermoelastic) mechanisms. These ideas were realized in experiments. It was determined that in the case of sound generation in water by a proton beam with the characteristics described above and broad-band detection of generated acoustic signals, the signal amplitude vanishes at 6°C (Fig. 10.51), but in the case of detection in a relatively narrow band corresponding to the energy-carrying band of N-wave produced due to the thermoradiation mechanism (this band is determined by the time of sound travelling along the transverse dimensions of a beam), the acoustic signal vanishes at a temperature of 4°C (Fig. 10.52)
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which corresponds to the temperature dependence of the coefficient of thermal expansion of water (Fig. 10.53).
Figure 10.49 Dependence of coefficient of thermal expansion α (a solid line) and acoustic signal amplitude (circles) on temperature [253].
Very interesting studies of sound excitation by a proton beam in water were conducted by Danil’chenko et al. [84]. Measurements were performed using a beam from the synchrotron of the Institute of Theoretical and Experimental Physics (Russia). Protons with energy 200 and 190 MeV were used. The collimator diameter was 4 cm. Water temperature was monitored. The pulse length was 100 ns and much smaller than the length of acoustic pulses (τa ≤ 100 µs). This made it possible to consider the process of heating of water volume in the region of beam action to be instant. Total energy release was varied within the range 8⋅1015 ÷ 23⋅1019 eV. Measurements of acoustic signals were performed by a hydrophone and a measuring circuit with the frequency band 0.1 ÷ 80 kHz in the near wave field of a thermoradiation sound source produced in water by the beam. The amplitude of the positive half-wave of N-signal was detected. Figure 10.54 shows the dependence of the acoustic signal on the energy of protons, which was obtained as the result of the measurements. Data by other researchers who conducted analogous measurements using proton, laser,
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and electron beams of various energies [68, 69, 254] are given in this figure for comparison.
Figure 10.50 Dependence of acoustic signal amplitude on temperature [253].
Figure 10.51 Shape of acoustic signal at various values of temperature in the frequency band 7 – 80 kHz [220].
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Figure 10.52 Shape of acoustic signal at various values of temperature in the frequency band 7 – 40 kHz.
Figure 10.53 Dependence of acoustic signal amplitude on temperature in the frequency band 7 – 40 Hz [220].
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Figure 10.54 Dependence of acoustic signal amplitude on particle energy. (1) Data by Danil’chenko et al. [84]; (2) data by Golubnichii, Kalyuzhnyi, and Korchikov [68]; (3) data by Golubnichii et al. [69]; and (4) data by Sulac et al. [254].
Figure 10.55 shows the dependence of acoustic signal amplitude on the hydrophone position at the beam axis [84]. The mechanism of proton absorption corresponding to the so-called Bragg peak is observed clearly at the end of the track of protons. The presence of this mechanism is caused by the known dependence of ionization losses on proton energy. The dependence of losses of α-particles of polonium in air on the residual track is given in Fig. 10.56 for comparison (see [188], p. 207). One can see a good correlation of results given in Figs. 10.55 and 10.56. The experimental results given above, which characterize sound generation by protons in a condensed medium, concern basically the case when the dimensions of medium volume are large in comparison with the track length of protons in a medium. However, the cases are interesting when the ratio of the track length to the target thickness changes within the range of change of proton energy. Such experiments were conducted by Borshkovskii, Volovik, and Lazurik-El’futsin [36, 38]. As it follows from physical concepts and the theory, a peak must be observed in the curve characterizing the dependence of acoustic signal amplitude on the energy of protons incident at a target shaped as a plate. The authors observed the peak and named it the acoustical peak of protons. The physical nature of this peak is essentially the same as that of the Bragg peak at the curve characterizing proton absorption in a medium. The experiments [38] were conducted using the accelerator of the Institute of Theoretical and Experimental Physics. Relatively low-frequency acoustic vibrations (f = 66 kHz) arising in an aluminum plate-target (the plate thickness h = 0.2 cm) were detected. Proton pulses (the pulse length τ = 20 µs) with initial energy Ep = 24.6 MeV in a beam with diameter
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d = 0.75 cm after a collimator were incident at the plate center. The energy of the proton beam was changed by transmitting it through a holder with decelerating plates made of copper foil.
Figure 10.55 Dependence of signal amplitude on hydrophone position along the beam axis [84].
Figure 10.56 Dependence of ionization created by α-particles of polonium on residual track [188].
Figure 10.57 shows the dependence of the displacements x1 (curve 1) and x2 (curve 2) of the front and rear surfaces of the plate, respectively on proton energy. Curves 1 and 2 were obtained by calculation on the basis of the simplest one-dimensional model of thermoelastic (thermoradiation) excitation of sound by protons in a plate. Circles indicate experimental results. Light circles correspond to the experiments by Borshkovskii, Volovik, and Lazurik-El’futsin [38]. The peak in curve 2 corresponds to the energy of protons with a track length in aluminum equal to the plate
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thickness (Ep = 10 MeV). A dashed-line curve corresponds to the half-sum of the displacements x1 and x2, which is proportional to absorbed proton energy according to estimations [38]. Black circles show experimental results published earlier by Borshkovskii and Volovik [36], which may be an illustration demonstrating ionization losses in the process of interaction of a proton beam with a substance and increase of the amplitude of excited sound. These experimental data are given in Fig. 10.58 presenting the dependence of the amplitude of the acoustic signal generated in an aluminum plate by protons with initial energy ~70 MeV on the thickness of a plexiglas plate (radiation absorber). Absorbing plates were installed in the path of a proton beam in the front of the aluminum plate. The experiments were conducted using the linear accelerator of the Institute of Technical Physics of the Academy of Sciences of Ukraine. The maximum value of the amplitude of acoustic signal corresponds to the Bragg peak. Results of calculation corresponding to these experimental data are given in Fig. 10.57. These experiments [38, 175] demonstrated that the maximum corresponds to such energy of protons incident on an aluminium target cell, when the track length of protons in the aluminium plate is equal to its thickness. The results [36, 38] agree with the theoretical conclusions given in Chapter 6: the condition of equality of the radiation length to the plate thickness is optimal for sound generation by penetrating radiation.
Figure 10.57 Dependence of acoustic signal amplitude on proton energy for an aluminium plate [38].
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Figure 10.58. Dependence of signal on the thickness of a plexiglas absorber [38].
8. EXCITATION OF ACOUSTIC WAVES IN METALS BY ELECTRONS, POSITRONS, AND γ-QUANTA Sound excitation by electrons was considered in the first papers on radiation acoustics. At first it was a theory describing generation of the Cherenkov acoustic radiation by an electron moving uniformly with a supersonic velocity in a metal [101, 203]. The first papers devoted to radiation-acoustic experiments also concerned sound excitation by an electron beam in solids. It was characteristic that these studies were conducted almost simultaneously with the first experiments on laser excitation of acoustic waves in solids [260, 261]. Multiple experiments on sound excitation by electrons in condensed media including sound excitation by electron beams in metals were performed after that. These studies were conducted using various accelerators in the broad energy range in conditions, where the thickness of sample-targets was larger than the radiation length, or with thin plates (see [37, 39, 197, 242] for example). Excitation of acoustic waves by electron beams in water was investigated by Golubnichii et al. [69] and Lyamshev and Chelnokov [156]. Papers discussing the nature of acoustic waves generated by electron beams in crystals and natural solids were published recently [3, 15, 45, 116, 170]. We will give some typical results of studies of excitation of acoustic waves by electrons, positrons, and γ-quanta in metals. Borshkovskii et al. [37] experimented with the linear accelerators of the Physical and Technical Institute of the Russian Academy of Sciences. The scheme of experiments was traditional. A beam of electrons (positrons) from a linear accelerator was incident on a plate made of the metal under investigation. A piezoelectric transducer was installed on the plate. The transducer was
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connected to an amplification and detection circuit. A typical length of a particle “package” incident on a sample-target was τ = 1 µs. A flow of bremsstrahlung was produced by an electron beam with energy Ee = 620 MeV in a tantalum target with thickness 5.3⋅10–2 cm. After transmission through a series of collimators, the beam of bremsstrahlung had an average transverse dimension d = 1.5 cm at the surface of the studied target plate. Figure 10.59 shows the dependence of the amplitude of acoustic signal on the diameter of an electron beam at the target, a lead plate with the thickness 0.2 cm. The plate was thin, i.e., its thickness was smaller than the track length of electrons. The experiments were conducted at electron energy Ee = 20 MeV, a constant number of particles in the beam, and radiation pulse length τ = 2 µs. Thus, in the case of variation of the beam diameter at the plate surface, the total energy (power) of the electron beam stayed constant. One can see that the amplitude of acoustic signal changes approximately in inverse proportion to the square of the beam diameter. If we take into account the fact that the efficiency of thermoradiation conversion is directly proportional to the intensity of penetrating radiation, then these experimental data agree well with the results of the theory of thermoradiation sound excitation in solids as in the cases of sound excitation by beams of protons and synchrotron radiation (X-rays) considered earlier.
Figure 10.59 Dependence of acoustic signal amplitude on the diameter of an electron beam at a target [37].
Figure 10.60 demonstrates the results of measurements of the dependence of the amplitude of acoustic signal in the case of a thick aluminium plate (h = 5 cm). The plate thickness in these experiments was of the order of magnitude of the track length of electrons in aluminium, and the length of the radiation pulse was smaller than the propagation time of sound along the beam dimensions in the plate. The observed amplitude of acoustic signal changes directly proportionally to the electron energy in the beam, as follows from theory.
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Figure 10.60 Dependence of acoustic signal amplitude on the energy of electrons for a thick aluminium plate [37].
Figure 10.61 Dependence of acoustic signal amplitude on the number of electrons for a thick lead plate [37].
A dependence of acoustic signal amplitude in a thick lead plate on the number of electrons (positrons) in a pulse of penetrating radiation is shown in Fig. 10.61. The plate thickness was h = 5.0 cm. The particle energy was Ee = 620 MeV. As the total energy in a radiation pulse is directly proportional to the number of particles, the experimental results agree with theory. Figure 10.62 gives the dependence of acoustic signal amplitude on the energy of electrons for a thin aluminium plate (h = 0.2 cm). In the considered case the plate thickness is small as against the radiation length. The energy of electrons in this experiment was smaller than the critical one, i.e., Ecr ≤ 40 MeV, and the main losses in the process of absorption of an electron beam in aluminum were ionization losses. In this case the total number of secondary electrons or δ-electrons in the energy range from 10 to 30 MeV did not change. Just this fact is the explanation of the absence of dependence of acoustic signal amplitude on the energy of electrons in a beam.
Figure 10.62 Dependence of acoustic signal amplitude on the energy of electrons for a thin aluminium plate [37].
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Finally, the dependence of intensity (not amplitude!) of acoustic oscillations excited in a thick lead plate on the total number of equivalent photons of γ-radiation at Ee = 620 MeV is given in Fig. 10.63. One can see that in this case direct proportionality between the amplitude of acoustic signal and the energy of photons of γ-radiation is also observed.
Figure 10.63 Dependence of acoustic signal intensity on the total number of equivalent photons for a lead plate with the thickness 5.0 cm [37].
Malugin and Manukin [161] give data on sound excitation in solids by a beam of low-energy electrons. These data deserve our attention also because of the fact that the target was an aluminium cylinder with the mass 1 t. The cylinder had the first quadrupole mode of vibrations 104 rad/s and the Q-factor 105. Cylinder vibrations were detected by a capacitance transducer. Signal accumulation was performed. The source of electrons was an electron gun positioned near the lateral surface of the cylinder. The gun accelerated electrons to an energy of 0.5 keV. The length of a current pulse was τ = 8 µs. The interest in investigation of the impact of low-energy particles on such massive bodies arose in connection with the problem of detection of gravitational waves with the help of detectors in the form of massive elastic bodies. Figure 10.64 shows the dependence of the amplitude of acoustic vibrations in a cylinder on the energy of electron beam. Considering given data, one can draw the conclusion (as Malugin and Manukin did themselves [161]) of direct proportionality of an acoustic signal to the energy of electrons. Thus, the experimental data given are the evidence in favor of the thermoradiation mechanism of sound generation by beams of electrons (positrons) and γ-quanta and agree with the theory presented in previous sections.
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Figure 10.64 Dependence of amplitude of acoustic vibrations in a cylinder on the energy of an electron beam [161].
9. SOUND GENERATION BY AN ELECTRON BEAM IN WATER In conclusion of this chapter we give the experimental results obtained by Lyamshev and Chelnokov [156], who investigated sound excitation by an electron beam in water. Experiments were conducted at the linear accelerator of electrons of the Nuclear Research Institute of the Russian Academy of Sciences. This accelerator provided an opportunity to obtain a beam of accelerated electrons with energy in the range from 10 to 70 MeV. The length of pulse of the electron beam was 1.4 µs. The modes of a single pulse of electron beam and a sequence of pulses with repetition frequency 50 Hz were possible. The average current of an electron beam in the mode of a sequence of pulses with the repetition frequency 50 Hz could be changed from 0 to 10 µA, i.e., the energy in a single pulse could be changed from 0 to 10 J. Figure 10.65 gives the scheme of the experimental setup used for investigation of sound generation in water by an electron beam. The setup consists of an accelerator of electrons LUE-100, a special basin with acoustically insulated walls, a wide-band hydrophone and a wide-band amplifier, a cable connecting the control room with the physical room, and a system of detection of acoustic signals. The basin was equipped with a coordinate device for hydrophone positioning. An electron beam was set with respect to the current strength, electron energy, and shape of the
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section of an acoustic signal. A layer of a special rubber with reflection coefficient at frequencies over 5 kHz not smaller than 0.2 (with respect to pressure) was fixed at the internal surface of the basin walls. A special window was installed in the place of the beam entry into the basin. The window was covered by a polyethylene disk. The basin itself was installed upon a layer of foam plastic in order to reduce possible acoustic coupling of basin walls with its mounting and the collimator. The wide-band hydrophone of 5-mm diameter had an average sensitivity 10 µV/Pa and a band up to 200 kHz. The amplifier provided amplification of about 1.7⋅103 with nonuniformity not larger than 6 dB in the same band. Detection was performed by photographing oscilloscope scans. Oscilloscope scans were triggered by the noise pulse accompanying the startup of the electron beam. It was determined that the length of the noise pulse triggering the oscilloscope scan was about 40 µs. Therefore, there was no sense in positioning the hydrophone closer than 6 cm from the region of absorption of the electron beam (the region of sound generation). The shape of the thermoacoustic array produced in water, was determined after this with the help of measurements of delay times from the scan start at the oscilloscope screen to the starting time of an acoustic signal at various positions of the hydrophone in the basin. The thermal mechanism of sound generation was confirmed by both proportionality of acoustic signal amplitude to the total energy contained in a pulse of electron beam and significant decrease of acoustic signal in the case of pulldown of water temperature to 4°C.
Figure 10.65 Scheme of experimental set-up. (1) A hydrophone; (2) an amplifier; (3) a cable; (4) an oscilloscope with a camera; (5) a basin; (6) a layer of foam plastic; (7) rubber walls; (8) an input acoustic decoupling; (9) a collimator; (10) a film for monitoring of transverse dimensions of a beam; and (11) accelerator output.
It is necessary to note that significant heating of the region of beam absorption was observed in the mode of sequence of pulses with repetition
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frequency 50 Hz under water temperature close to 4°C (the acoustic signal increased with time). The same effect under room temperature (and more precisely at 16°C) manifested itself to a much less extent (it was almost imperceptible). This is quite natural if we proceed from the fact that the electron beam heats the region of absorption for several kelvins, heating from 4°C for several kelvins giving much larger increase of acoustic signal than heating for the same amount of kelvins from 16°C. This happens because of the fact that the coefficient of thermal expansion of water is close to zero at a temperature of 4°C and increases almost linearly with further increase of temperature. Figure 10.66 gives the sectional view of the basin with a thermoacoustic array formed in it for two values of electron energy in the beam: Ee = 20 and 50 MeV. The shape of this array represents a truncated cone with the opening angle of about 10°. The length of this array is about 10 cm for electrons with energy Ee = 20 eV and about 25 cm for electrons with the energy Ee = 50 eV which corresponds to the energy loss by an electron equal to 2 MeV per 1 cm track in water.
Figure 10.66 Sectional view of basin and thermoradiation array formed in water at electron energy (1) 20 and (2) 50 MeV.
If the hydrophone position with respect to the thermoacoustic array is changed, the shape and amplitude of acoustic signals change too. The shape and amplitude of acoustic pulses in the case of the hydrophone position with respect to the axis of the thermoacoustic array at a distance of 10.5 cm and different distances from the array origin is given in Fig. 10.67. In this case the energy of electrons is equal to 20 MeV and the average current is 2.5 µA, such that the energy in a single pulse is about 1 J. The initial diameter of the electron beam at its entry in the basin window is equal to 2 cm. If we position the hydrophone near the origin of the thermoacoustic array (curve 1) then this corresponds to the approximation of the near wave field from an instantly heated cylinder. Sound pressure in a liquid in this case can be presented in the form
αE c p= L πR
∞
ω 2 a 2 π − cos − ω R − t dω , ω exp ∫ 4c 2 4 c 0
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where E is the total energy of a pulse of electron beam, L is the length of a thermoacoustic array, R is the distance form the array axis, and a is the beam radius. In this case for curve 1 in Fig. 10.67, the calculated value of amplitude is equal to 40 Pa and the duration of the compression phase is about 20 µs, which agrees well with the experimental values of 30 Pa and 22 µs. If the distance from the hydrophone to the origin of the thermoacoustic array is increased, the approximation of the expression given above ceases to be valid in particular, because of the finiteness of the array size. Moreover, there is a possibility of influence of waves reflected from the polyethylene disk covering the input window of the basin and direct emission of acoustic waves by the disk itself on the shape of acoustic pulses.
Figure 10.67 Shape and amplitude of acoustic signals in the case of the hydrophone position at a distance 10.5 cm from the axis of the thermoradiation source and different distances from it: 3, 8, 10.5, 13, and 18 cm (curves 1 – 5, respectively).
An interesting detail was noted. When the electron beam hit directly a basin wall (not through the special window), the character of the acoustic signal changed sharply: its amplitude increased several times and its initial shape changed strongly. This result may be confirmation of the theoretical conclusion on the possibility of significant change of an acoustic signal in a multi-layer medium. The experiments conducted provide an opportunity to draw a conclusion on the fact that in the cases, where theoretical results (based on the thermal
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mechanism) can be compared to experimental data, good agreement is observed.
10. SOUND EXCITATION BY A BEAM OF IONS IN METALS Papers devoted to sound excitation by ion beams in metals were published recently [189, 251]. The results of experimental studies of generation of + acoustic waves by a beam of Ar ions in aluminium published by Satkewicz et al. [251] may be the most characteristic. An aluminium disk with diameter 1.4 cm and thickness 0.3 cm was used in experiments. A piezoelectric detector made of PZT-ceramics was fixed at one side of the + disk. A beam of Ar ions was incident on the opposite side of the disk. Their energy changed within the range from 1 to 10 keV and current in the circuit “beam – target – piezoelectric detector” could be changed from 0.3 to 14 µA at constant ion energy (in other words, the number of particles in the beam was changed). The modulation frequency of ion intensity in the beam could be changed also from 15 Hz to 20 kHz. The diameter of the + beam of Ar ions at the target surface (an aluminium disk) was equal to 300 µm. Some results of the experiments are given below.
Figure 10.68 Dependence of acoustic signal amplitude in aluminium on current in a + beam of Ar ions at different values of particle energy in the beam [251].
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Figure 10.68 shows the dependence of the amplitude of an acoustic signal at the output of the piezoelectric detector on the current in the beam at the frequency of modulation of its intensity 2 kHz. The parameter is the energy of ions in the beam. The dependence of acoustic signal amplitude S b on the current ip can be approximated by an expression S ≡ aip , where b is the index characterizing the inclination of the straight line and equal to 0.96 approximately, and a is the numerical coefficient. Analogous dependences were observed within the whole range of changing of the modulation frequency of beam intensity from 15 Hz to 20 kHz. The data characterizing the dependence of acoustic signal amplitude at the output of the piezoelectric detector on the voltage Vi characterizing ion energy are given in Fig. 10.69. The observed dependences may be described b by the relationship S = aVi as above. The index value here is also equal to b = 0.96 approximately.
Figure 10.69 Dependence of acoustic signal amplitude in aluminum on the energy of an ion beam at different values of current in the beam [251].
These experiments demonstrate that an almost linear bond between the amplitude of an acoustic signal and the energy (or the number of particles) of ions. The authors explain a certain small difference from linear dependence (the index is not equal to one still) by the fact that the mechanism of direct transfer of momentum of particles in the beam to the metal target takes part in generation of acoustic waves by heavy ions in metals apart from the thermoradiation and thermoelastic mechanisms. The role of the last mechanism becomes less and less significant as the particle energy grows and therefore, in the case of moderate density of energy of
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penetrating radiation absorbed in a substance and large particle (ion) energy, the dominant mechanism is the thermoradiation mechanism.
CHAPTER 11
Some Applications of Radiation Acoustics Investigation of acoustic effects of interaction of penetrating radiation with matter opens new opportunities for investigation of penetrating radiation itself (acoustic detection and radiation-acoustic dosimetry and diagnostics), study of physical characteristics of substances, nondestructive testing (radiation-acoustic microscopy and radiation-acoustic sounding and visualization of inhomogeneous condensed media), and also nontraditional ways of controlled radiation-acoustic action upon physical-mechanical and chemical structure of substances. Accelerators and lasers are being introduced to a larger and larger extent in modern technology. Unification of radiation, laser, and ultrasonic technologies provides basically new opportunities for solution of important applied problems. Some of applications of radiation acoustics have been described in literature already (see [97] for example). Here we give only the examples of its applications concerning monitoring of product quality (radiation-acoustic microscopy and visualization), detection of super-high-energy elementary particles, and also some applications of new-generation super-powerful accelerators, which could seem very unusual.
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1. SCANNING RADIATION-ACOUSTIC MICROSCOPY AND VISUALIZATION Traditional techniques of investigation and visualization of microscopic objects such as optical and electron microscopy have many restrictions. For example, an optical microscope and a scanning electron microscope have high resolution but they are unsuitable for investigation of internal regions of opaque materials. If X-ray TV microscopes are used, difficulties connected with interpretation of the images obtained arise. This is especially true in the case of studying low-contrast objects. Radiation-acoustic microscopes do not have such shortcomings [154, 157]. The action of a radiation-acoustic microscope is based upon the phenomenon of generation and propagation of sound and thermal waves in an object, which are excited by the sounding intensity-modulated penetrating radiation. It should be noted that in the majority of cases, the role of heat conductivity is ignored in the process of consideration of sound generation by penetrating radiation since the dimensions of the region of heat release are always large compared with the thermal wavelength. On the contrary, in the case of a radiation-acoustic microscope, a beam of penetrating radiation is focused, the dimensions of the region of heat release are small, and thermal waves often play a fundamental role. Acoustic vibrations and thermal waves arising in an object are detected by sound detectors most frequently. An acoustic signal depends on local physical properties of an object. Therefore, in the case of scanning by a beam in two mutually orthogonal directions, a radiation-acoustic image of an object is formed. In the general case it results from three processes: variation of absorbed power of penetrating radiation due to change of radiation properties of an object from one point to another, interaction of thermal waves with thermal inhomogeneities of an object, and interaction of acoustic waves with elastic inhomogeneities of an object. The first process provides information on only the radiation-absorption properties of an object. If this process is dominant, the radiation-acoustic image is essentially identical to the optical or scanned electron image. The resolution of the radiation-acoustic microscope in this case is determined by the diameter of the sounding beam, and the depth of visualization of a subsurface structure is determined by the penetration depth of radiation. The second process is characterized by interaction of thermal waves with microscopic inhomogeneities of an object. It gives qualitatively new information and provides an opportunity to expand essentially the knowledge on physical properties of an object. The third process carrying information on mechanical irregularities of an object plays an essential role if the acoustic wavelength is of the same order of magnitude as the dimensions of microscopic inhomogeneities in an object (usually this takes
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place at modulation frequencies of penetrating radiation exceeding 100 MHz). In this case the radiation-acoustic image is identical to the acoustic one (as in an acoustical microscope) and resolution is of the order of magnitude of the sound (hypersound) wavelength.
2. SCANNING LASER-ACOUSTIC MICROSCOPY Historically, the first example of radiation-acoustic microscopy was laseracoustic microscopy (or photoacoustic microscopy as it is often called) [162, 205]. A typical block scheme of a photoacoustic microscope is shown in Fig. 11.1. An intensity-modulated laser beam (of the infrared, ultraviolet, or visible range) scans the surface of an object under investigation. Modulation is performed by mechanical or electrooptical methods. An acoustic signal from a detector is fed to a synchronous detector via a preamplifier. The output of the synchronous detector is connected to a visualization device (a display, plotter, or storage oscilloscope) with the scans synchronized to the system of scanning of a laser beam. Depending on the way of detection of acoustic signals, photoacoustic microscopes are divided into schemes with microphones and schemes with piezoelectric transducers. There are also schemes of photoacoustic microscopes with detection of a useful signal with the help of an auxiliary laser beam or a photodetector.
Figure 11.1 Block scheme of a scanning photoacoustic microscope. (1) A laser, (2) a modulator, (3) a control system of scanning, (4) a focusing system, (5) an object under investigation, (6) an acoustic (thermal) detector (sensor), (7) a preamplifier (in the case of harmonically modulated laser radiation it is an amplitude or phase detector and in the case of pulsed laser radiation it is a spectrum analyzer), (8) a synchronous detector, (9) and (10) scan generators, and (11) a visualization device.
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In the case of a microphone technique a sample is placed into an optoacoustic cell (Fig. 11.2) consisting of the hermetic chamber filled with a gas or air, a microphone, and a sample holder. The chamber has a window transparent to sounding laser radiation. Acoustic oscillations arising in a gas chamber under the effect of a laser beam on an object are detected by a sensitive microphone. In the case of a photoacoustic microscope with a piezoelectric transducer (Fig. 11.3), a studied object is in direct contact with the piezoelectric transducer detecting bulk acoustic waves. In the case of a photoacoustic microscope with detection of an optoacoustic signal by an auxiliary beam (Fig. 11.4), either the change of the optical refraction coefficient in a medium in the layer near an object or sound vibrations of an object are detected. The last version of detection is especially convenient in the case of investigation of surfaces with protrusions and cavities.
Figure 11.2 Block scheme of an acoustic (gas-microphone) cell. (1) An input window, (2) a chamber wall, (3) an object holder, (4) a microphone, and (5) an object.
Figure 11.3 Receiving part of a photoacoustic microscope with a piezoelectric transducer. (1) An object, (2) a transducer, and (3) an object holder.
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Figure 11.4 Method of detection with the help of an auxiliary laser beam. (1) A sounding laser beam, (2) a heated region of an object, (3) an object, (4) a direction of mechanical scanning, (5) an auxiliary laser, and (6) a photodetector.
Figure 11.5 Optoacoustic (photoacoustic) image of an electronic chip (an argon laser with the power 0.1 W; the modulation frequency of light intensity 1 kHz; the resolution of photoacoustic microscope 5 µm).
Some fields of application of photoacoustic microscopy are nondestructive profile analysis, i.e., investigation of structure of layered inhomogeneous materials; study of electronic chips (Fig. 11.5); monitoring of chemical composition of complex chemical compounds; investigation of the crystal structure of semiconductors in the process of ion implantation; an opportunity to visualize volumetric or surface areas with different
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thermal characteristics because of inhomogeneity of crystal structure; direct monitoring of laser annealing; study of phase transitions in crystals; and also measurements of thickness and monitoring of uniformity of anodic deposition of films upon semiconductor substrates. Designers of photoacoustic microscopes expect a lot from applications not only in electronic industry but also in medical sciences and biology. Photoacoustic microscopes are inferior to optical and electron microscopes in resolution but surpass them in the amount of information contained in images as they provide an opportunity to visualize the details of microstructure of objects opaque to photons and electrons, open new fields of microscopy, and may broaden essentially traditional techniques of microscopic analysis.
3. SCANNING ELECTRON-ACOUSTIC MICROSCOPY Schemes of photoacoustic microscopes with electronic excitation, where the role of the laser beam is performed by an electron beam [208], have been developed and used during the last decades. A focused electron beam is used for excitation of acoustic and thermal waves in a sample (solid) and acoustic signals are detected by a piezoelectric detector being in direct contact with a sample. In other words, the same scheme as in a photoacoustic microscope with piezoelectric detection is used. The first papers describing schemes of electron-acoustic microscopy were published more than ten years ago (in 1980) [201, 207]. The first scanning electronacoustic microscope was described in 1983 [201, 207, 211]. The advantage of utilization of electrons instead of photons is first of all the fact that an electron beam can be focused into a spot of smaller dimensions. Moreover, the track length of electrons (le = µ −1) in media opaque to light can be essentially larger than that of photons. Both these factors open opportunities to increase the resolution of microscopes. Estimations show that the value of le in solids can be determined with a precision sufficient for applications according to a formula [243] l e = E e1.43 / 10 ρ ,
where le is in micrometers, Ee is in keV, and ρ is in g/cm3. For example, in the case of the electron energy Ee = 30 keV, we obtain le = 0.7 µm for gold, le = 1.4 µm for copper, and le = 5.5 µm for aluminum. The first scanning electron-acoustic microscopes were designed on the basis of standard scanning electron microscopes. They were a certain
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version of their modification. Essentially, a standard scanning electron microscope was complemented just with several devices: a modulator of electron beam intensity (additional deflector plates in the chamber of a scanning electron microscope) and a sample holder equipped with a piezoelectric detector with electronic circuitry necessary for amplification and visualization of an acoustic signal. Scanning electron-acoustic microscopes turn out to be more universal devices than standard scanning electron microscopes or even scanning photoacoustic microscopes. Moreover, one and the same device can operate in both modes (the modes of a scanning electron-acoustic microscope and scanning electron microscope) as a rule. Comparison of images obtained in different operation modes provides new opportunities in investigation of the structure of a studied sample object.
Figure 11.6 Scheme of electron-acoustic visualization [208]. (1) A focused electron beam (with modulated intensity), (2) a periodically heated area performing periodical expansion, (3) ultrasonic waves generated by the periodically expanding area in the sample, (4) a piezoelectric detector, (5) electron beam scan, (6) the output of a piezoelectric detector used to enlarge images, and (7) changes in brightness of an enlarged image, which indicate conversion of electron energy into an acoustic signal depending on the properties of sample material.
Figures 11.6 and 11.7 show schematically the major processes of image formation and the block scheme of a scanning electron-acoustic microscope designed on the basis of a standard scanning electron microscope. The following data can give some idea on its parameters. In the case of the accelerating voltage 30 kV, the peak value of current in an electron beam with spot size of about 1 µm attains 10 µA. Successful experiments were
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reported with an electron beam with diameter at the sample surface of about 0.1 µm and maximum (peak) current in it 0.1 µA. A piezoelectric detector and a sample holder are combined quite often in a single device. For example in one of the versions of scanning electronacoustic microscopes, a sample was glued to a piezoelectric detector (a disk made of PZT-ceramics with thickness 0.5 mm and diameter 12 mm).
Figure 11.7 Block scheme of a scanning electron-acoustic microscope on the basis of a standard scanning electron microscope [208]. (1) The chamber of a scanning electron microscope, (2) an electron gun; (3) the first electronic lens, (4) the second electronic lens, (5) deflector plates, (6) the terminal lens and yokes, (7) the terminal aperture, (8) a sample and a piezoelectric detector, (9) a generator of pulsed and sinusoidal signals, (10) a rectifier, (11) an amplifier, (12) a detector, (13) an amplifier, and (14) an imaging device (a TV tube).
The optimal rate of electron beam scanning over the sample surface in a scanning electron-acoustic microscope depends on time constants of the electronic devices and the frequency of beam modulation. If the diameter of the element to be resolved at the sample surface is about 1 µm and the scan area at the sample under investigation is 0.1 × 0.1 mm2, there are
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0.1/0.001 = 100 elements to be resolved along only one scan line. If we take 1000 cycles-oscillations to resolve a single element in the case of modulation frequency of beam intensity 1 MHz and 100 lines in the image, visualization of a single electronic (acoustic) image takes 10 s. In the case of slow scanning the signal-to-noise ratio is improved and the resolution of a scanning electron-acoustic microscope increases.
Figure 11.8 Image of a silicon structure with a phosphorescent coating [208].
Scanning electron-acoustic microscopy and visualization are used in microelectronics to monitor the quality of electronic chips, determine
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defects of the crystal structure of metals and alloys, study the character of dislocations and other defects in material characteristics under large loading, visualize vibration modes of surfaces, and monitor nondestructively the presence of defects (cracks) in microscopic samples, etc.
Figure 11.9 Image of structure of borders in polycrystalline copper obtained by a scanning electron-acoustic microscope [208].
Figure 11.8 shows images of a silicon structure with a phosphorescent coating in order to illustrate the opportunities provided by scanning electron-acoustic microscopy. One can see that in the image obtained with the help of a standard scanning electron microscope (Fig. 11.8a), silicon structures under the coating are invisible. On the contrary, Figs. 11.8b and d demonstrate a good quality image of a silicon structure obtained with the help of a scanning electron-acoustic microscope. An image of the structure of crystalline particles in a polycrystalline copper sample obtained by a scanning electron-acoustic microscope is given in Fig. 11.9. Characteristic changes of image brightness are visible directly at the borders.
4. X-RAY – ACOUSTIC SCANNING VISUALIZATION Suggestions to use a beam of X-rays instead of a laser beam in photoacoustic microscopy and visualization were discussed by Kim, Sachse, and Pierce [226, 250]. These authors published the first experimental results of studies of scanning X-ray – acoustic visualization [250]. They used the synchrotron radiation of the X-ray range with energy 10 keV. The source
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was the high-energy synchrotron source of Cornell University (USA). The radiation represents a periodic sequence of pulses with length 0.160 ns, energy 1.12 µJ per pulse, and repetition frequency 390.6 kHz. The studies were conducted using aluminium sample disks with diameter 5.72 cm and thickness 1.57 cm. It is necessary to note that the track length of X-ray quanta with energy 10 keV in aluminium is 0.14 mm. In the case of laser radiation, it equals 3.76 nm.
Figure 11.10 Results of measurements of spatial distribution of mean-square values of amplitudes of acoustic signals excited by a modulated X-ray beam [250].
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A series of experiments was conducted in the case of scanning by the beam in mutually perpendicular directions along the sample surface. The beam dimensions at the sample surface were 2 × 2 mm2 and the dimensions of the scanned area were 12 × 12 mm2 or 18 × 18 mm2 with orientation according to the center of a disk-target. An acoustic signal was detected by a piezoelectric transducer made of PZT-ceramics. Detectors with diameters 18.5 and 1.3 mm were used. The detectors were fixed in the center of a sample disk at the surface opposite to the irradiated surface. Experiments with a detector of 1.3-mm diameter fixed at the lateral side of the disk were conducted also. Detectors were damped in order to secure the necessary frequency band. The studies were conducted using two modes. In one of them a collimated beam of X-rays was directly applied to the surface of a disk sample and moved in mutually perpendicular directions. The mean-square value of acoustic signal amplitude was detected at the output of the detection-amplification circuit. This value was represented at a video monitor synchronously with the beam motion. In the other mode called by the authors the mode of double modulation, a low-frequency intensity modulation of X-rays that already were a periodic sequence of pulses with a very high repetition rate (two or three orders of magnitude higher than the modulation frequency) equal to 390.6 kHz was performed. A mechanical modulator in the form of a disk chopper made of stainless steel was installed after a collimator in front of the sample target. Modulation of the initial flow of X-rays with frequency from 0.5 to 2.5 kHz was performed at certain rotation rates of the disk chopper. The amplitude and phase of the envelope of a low-frequency acoustic signal could be detected at the receiver output. Utilization of the mode of double modulation was based on the understanding of the fact that the thermal wavelength in a sample target corresponding to the low-frequency envelope was larger than that for the repetition rate of the sequence of pulses of initial X-ray flow (λT ≡ f −1/2). This provides an opportunity to increase the depth of the monitored (by a thermal wave) subsurface layer of a sample (thermal wavelength). The last fact is important as acoustic waves of megahertz range decay rather rapidly. One of the basic problems, which the authors tried to solve, was to clarify the opportunities to monitor defects by an X-ray – acoustic technique. Experiments were conducted on visualization of a spatial pattern of distribution of amplitudes of acoustic signals excited by an X-ray beam in a disk sample with an internal inhomogeneity in the form of a cylindrical cavity oriented in the disk plane. Measurements conducted both without low-frequency modulation of X-rays and in the modulation mode did not give a satisfactory image of the cavity in the pattern of spatial distribution of amplitudes of acoustic signals excited in the disk by X-rays. It was suggested in this connection to use the ratio of mean-square values of
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amplitudes of acoustic signals of a low-frequency envelope in the target at two modulation frequencies as an informative signal. Experimental results are shown in Fig. 11.10. Figure 11.10a presents a pattern of spatial distribution of mean-square amplitudes of acoustic signals corresponding to the envelope at frequency 2.5 kHz and Fig. 11.10b corresponds to frequency 0.5 kHz. Figure 11.10c and d show the spatial distribution of the envelope of mean-square values of amplitudes of acoustic signals for the envelopes at frequencies 2.5/0.5 kHz and 1.5/0.5 kHz, respectively. One can see a rather clear image of the defect (cavity) inside the sample disk. The experiments conducted demonstrated a real opportunity to use X-ray – acoustic visualization for nondestructive testing. It is evident that there is a real opportunity to implement scanning X-ray – acoustic microscopy under the condition of sufficient focusing of an X-ray beam.
5. ION-ACOUSTIC MICROSCOPY AND VISUALIZATION Application of an ion beam to scanning radiation-acoustic microscopy and visualization was discussed for the first time apparently in 1983 by Lyamshev and Chelnokov [154, 157]. Design of an ion-acoustic microscope was reported in 1985 [257]. Results of experimental studies of scanning ionacoustic visualization of defects in metals were discussed by Satkewicz et + al. [251]. A beam of Ar ions was formed, directed, and moved at the surface of an aluminium sample with the help of modified ray optics of a standard mass-spectrometer. The beam formed a spot with diameter 300 nm at the sample surface. Accelerating voltage was varied within the range from 1 to 10 kV and the modulation frequency of the beam intensity was varied from 15 Hz to 20 kHz. The experiments pursued two goals. It was necessary to determine experimentally the character of the mechanism of radiation-acoustic conversion and clarify an opportunity to use it for ionacoustic visualization. It was determined experimentally that in the considered range of ion energy, the dominant mechanism of sound excitation by a modulated ion beam in the process of its interaction with a target is the thermoradiation mechanism. It was demonstrated also that an ion beam may be used effectively for scanning radiation-acoustic microscopy and visualization together with beams of photons (laser beams), electrons, and X-rays. Experiments were conducted using samples in the form of aluminium disks with diameter 1.4 cm and thickness 0.3 cm, which had cylindrical cavities with internal diameter of 0.1 cm inside.
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Figure 11.11 shows schematically a device for fixation of a sample and a piezoelectric detector used for detection of acoustic signals excited by ion beams in an aluminum sample.
Figure 11.11 Scheme of a device for fixation of a sample with a piezoelectric + detector [251]. (1) A beam of Ar ions, (2) an aluminium disk sample, (3) a cylindrical cavity simulating a defect, (4) PZT-ceramics, (5) direction of scanning, (6) an acoustic signal, (7) an insulating plate, (8) an earthing plate, and (9) sample holder.
Figure 11.12 Distribution of acoustic signal amplitude for an aluminum sample with two intersecting cavities.
Figure 11.12 demonstrates the change of an acoustic signal in the process of linear movement of an ion beam at the surface of an aluminum disk along a single line of the image and at different values of modulation
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frequency of beam intensity. An aluminum disk was used in experiments, which had two intersecting cylindrical cavities of 1-mm diameter inside at a depth D = 0.5 mm. One can see from the data given in Fig. 11.12 that at a low modulation frequency, when the condition λT/D > 1 (λT is the thermal wavelength) is satisfied, the amplitude of acoustic signal increases, when an ion beam is above the plane. On the contrary, if λT/D < 1, the signal decreases. At frequency 15 Hz the thermal wavelength in aluminium is equal to 1.45 mm and corresponds approximately to the distance between the centers of the cavities. The experiments [251] are also evidence of the fact that in the cases under consideration the resolution of ion-acoustic visualization at a high frequency is determined by the diameter (size) of an ion beam. Figure 11.13 gives the results of the experiments in which an aluminum sample disk with an inclined cylindrical cavity is scanned, i.e., a cavity with diameter 1 mm positioned at a certain angle with respect to the sample surface. The depth of cavity position was different in different places with respect to the sample surface where scanning was performed. As in the previous case, one can see that if the depth of cavity position is small (λT/D > 1), an increase of acoustic signal amplitude is observed. On the contrary, the signal amplitude decreases at λT/D < 1. Results given in Fig. 11.14 demonstrate the situation where a beam was moved at the surface of the same sample as in the previous case but along a fixed line, where the depth of cavity position was equal to D = 0.3 mm, and the modulation frequency of beam intensity was changed. Comparison of the results of measurements of acoustic signal amplitudes given in Figs. 11.13 and 11.14 indicates the existence of frequency dependence of the signal, which corresponds approximately to frequency law f −1/2 in the process of transition from the case λT/D > 1 to the case λT/D < 1. This is more evidence of the existence of two modes of visualization: a “thermal” mode, when thermal waves play the main role in formation of the acoustic signal, and an “elastic” mode, when the main role is played by elastic deformations, i.e., acoustic waves in a sample. Experiments on visualization of the same sample as in the case given in Fig. 11.14 but using scanning electron-acoustic microscopy and scanning laser acoustic microscopy are of interest too. Results of these experiments are shown in Fig. 11.15. The modulation frequency was about 78 kHz. This figure shows an image of a disk with a cavity obtained using a scanning electron-acoustic microscope and a line-by-line record of changes in the acoustic signal close to it (at the right). Corresponding records obtained with the help of a scanning laser-acoustic microscope are shown in Fig. 11.15b. One can see different contrast of images and the dominant roles of the “thermal” mechanism of radiation-acoustic visualization at λT/D > 1 and the “elastic” mechanism at λT/D < 1.
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Figure 11.13 Profile of an inclined cavity in a sample in the case of constant modulation frequency and variable depth [251].
Figure 11.14 Profile of an inclined cavity in the case of constant depth and variable modulation frequency [251].
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Figure 11.15 Image of an inclined cavity in an aluminium disk obtained using a scanning electron-acoustic microscope and laser excitation of sound [251].
The results given here (and in the preceding sections) indicate real opportunities to study various kinds of penetrating radiation in devices for radiation-acoustic visualization and microscopy.
6. ACOUSTIC DETECTION OF SUPER-HIGHENERGY PARTICLES IN COSMIC RAYS – THE DUMAND PROJECT The energy 1017 eV is commonly considered to belong to the range of super-high energy. The range boundary is not connected with a physical phenomenon. It is determined by the energy threshold of detection of extensive air showers by the largest existing installations such as those in Yakutsk (Russia) and Haverah-Park (Great Britain). The areas occupied by these two installations are 18 and 12 km2, respectively. Dimensions of installations are determined by the necessity to detect very rare events, i.e., emergence of particles of super-high energy in the spectrum of cosmic rays. The following data may give an idea of the number of detected events. During the decade of operation at Haverah-Park, they detected 70,000 showers with energy over 6⋅1016 eV, 52,000 showers with energy within the interval 1017 – 1018 eV, 4000 showers with energy over 1018 eV, 144 showers with energy 1019 eV, and only 16 showers with energy 5⋅1019 eV [22].
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Neutrino radiation arouses a lot of interest. This is caused by the enormous penetrating capability of neutrino. This capability provides a real opportunity to “look” into stars. This is especially true about studies at the energy Eν > 1 TeV, when only neutrino can carry information on highenergy processes taking place during the epoch with large red shifts, unique physical objects, i.e., hidden sources, etc. A new branch has arisen in astrophysics: neutrino astronomy. It is in the phase of its experimental development now. The Baksan neutrino telescope in the Northern Caucasus was put into operation. Similar installations are under construction in the USA and Italy. However, the main installation, which will open broad opportunities for experimental neutrino astronomy, is the DUMAND. It is in the design stage yet. We will discuss DUMAND below. Neutrino radiation is divided conventionally into two classes: atmospheric and cosmic neutrino radiation. In the first case neutrino are generated as a result of interaction of accelerated particles with atomic nuclei of matter in the atmosphere. Cosmic neutrino radiation arises in cosmic objects as the result of collision of accelerated particles with atomic nuclei and also due to the interaction of high-energy protons with lowenergy relict photons in space. Unstable particles like pions and kaons (π- and k-mesons) are born in the process of interaction of protons with atomic nuclei or photons. These particles disintegrate giving birth to muons (µ) and neutrinos (ν). High-energy neutrinos are detected by muons, hadrons, and electrons produced by them, which arise in the process of interaction of neutrinos with nucleons and give birth to nuclear-electromagnetic showers in an ambient medium or detector. In the case of high energy, a muon retains the direction of neutrino motion and has a large track length in soil or water. The track length of muons with energy higher than 1 TeV in water exceeds 3 km. Thus, muon detection provides an opportunity to determine “immediately” the direction of the neutrino source. In underground experiments mouns are registered by special detectors that are most frequently scintillation detectors. In underwater experiments the detector is water itself. Bremsstralung photons, electron-positron pairs, and hadrons giving rise to electromagnetic and nuclear-electromagnetic showers are created along a muon trajectory. At energy Eµ ≥ 100 TeV, an electromagnetic shower arises along a muon trajectory. Hadrons generated in neutrino-nucleon collisions give rise to nuclear-electromagnetic showers. The length of such a shower in water or ground is small in contrast to an electromagnetic shower generated by a muon. This is why the showers caused by a neutrino can be detected only if it interacted with nucleons inside a detector. In water the electrons of electromagnetic and nuclearelectromagnetic showers produce the Cherenkov optical radiation, which
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can be detected by a system of optical detectors, i.e., a lattice of photoreceivers. The idea of the possibility of recording cosmic neutrinos was suggested first by Markov in 1960 [239]. A new stage of high-energy neutrino astronomy began with the discussion of the project of the deep underwater experiment DUMAND (Deep Underwater Muon and Neutrino Detection). Initially the DUMAND detector was planned to be a spatial lattice of photodetectors submerged in the ocean at a depth of about 5 km. The distance between the sensitive elements (photodetectors) should not exceed the optical transparency length of water for visible and near-ultraviolet parts of the spectrum. The water layer over the installation serves as a shield from cosmic-ray muons. Askar’yan and Dolgoshein [8, 213] and independently Bowen [199] suggested an acoustic technique for detection of super-highenergy neutrinos (Eν ≥ 107 TeV) with the help of a spatial lattice of hydrophones. The essence of the technique of acoustic detection of super-high-energy neutrinos consists of the following. A nuclear-electromagnetic cascade produced by the interaction of a neutrino with the detector substance is accompanied by a fast (practically instantaneous) heating of water in a narrow channel along the shower axis. This causes the expansion of a volume of liquid in the channel and leads to a pressure pulse propagating in water perpendicularly to the shower axis. As estimates show, at very high energy, a narrow (several centimeters) and long (~10 m) particle beam forms in the vicinity of the shower axis. Heating of the medium within the region of action of the beam occurs due to ionization losses of the shower electrons slowed down to an energy lower than the critical one. Therefore, a cylindrical thermoradiation sound source forms in the region of absorption of shower electrons. Its radius (the radius of the heated part of the channel) is determined by the electron distribution over the channel cross-section. The length is L = [ln( Eh / Ecr )]1 / 2 , where Eh is the shower energy (the energy of hadrons initiating the shower), Ecr = 73 MeV is the electron critical energy, and L is in meters. We have given already the scheme of calculation of the sound field generated by a high-energy particle in a condensed medium and corresponding estimates in Chapter 7. However, it is expedient to give some estimates again. The authors of a large series of papers devoted to detection of high-energy particles (to the DUMAND project) analyze the parameters of the sound field in the near wave zone of an electromagnetic cascade arising in water, i.e., of the thermoradiation source of sound [9, 11, 56, 58,
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193, 194, 200]. They obtain the radiation acoustic pulse form and amplitude using the equation of thermoradiation sound generation in a liquid and calculating the energy release function Q(t, r) by different methods. The time dependence of the function Q(t) is always taken in the form of a δ(t)function as the initiation time of an electromagnetic cascade is much less than the other characteristic “acoustic” times, while the spatial dependence Q(r) is determined on the basis of some approximations and various direct calculations of the energy release density in the cascade. For example, the following expression for calculation of radiation-acoustic signal amplitude in sea water has been given by Askarijan et al. [194], p max =
0.44ϕ (r ) E r
1018
,
where p is the sound pressure (in Pa), E is the particle energy (in eV), and r is the distance from the channel axis (in meters). The factor ϕ(r) takes into account the deviation of the law of sound signal attenuation from the cylindrical one r−1/2 and ϕ(r) = 1.0; 0.95; 0.28; and 0.12 for r = 50, 100, 250, and 500 m, respectively. The acoustic signal amplitude for a particle with energy E = 1017 eV at a distance of 100 m from the cascade axis is equal to pmax = 4⋅10−3 Pa. The shape of an acoustic signal and its maximum amplitude depend, in fact, on the particular features of ionization losses in the nuclear-electromagnetic shower channel, i.e., on the form of the distribution function describing energy release Q(r). Figure 2.2 (Chapter 2) gives the shapes of acoustic pulses for different dependences of the function of energy release Q(r). In order to detect the acoustic signal from a shower in the ocean, it is important to know not only the signal amplitude and shape (its spectrum) but also the value of attenuation of sound during its propagation as well as interference characteristics, i.e., the spectrum of ambient noise in the ocean. All this leads to the necessity of determination of the optimal detection frequency band of the radiation-acoustic signal. The sound attenuation decreases as the frequency decreases (i.e., the sound absorption length increases) but, on the other hand, it is known that the level of ambient noise (interference) in the ocean increases as the frequency decreases. That is why in the majority of papers associated in one way or another with the DUMAND project, the estimates of the value of effective sound pressure are given either for some optimal frequency band or at a fixed (optimal) frequency. The opportunity to use the deposits of rock salt NaCl and Antarctic ice as the working medium of an acoustic neutrino detector was discussed. Basic opportunities for using various condensed media as the working medium of the detector were considered also.
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Effects of sound generation in condensed media by high-energy particles were considered explicitly in Chapter 7. Here we give only some estimates for the near wave field of a cascading particle. Let us recall the expression for the tensor of normal stress (sound pressure) in the case of observation of an acoustic signal in the direction perpendicular to the cascade axis, c 3 − 4 / n2 σ RR = − µE l 4πcε πr
2 2 ω a exp ω − ∫ cl2 0
∞
r π dω . cos t ω − − c 4 l
The parameters of the cascade formed by a high-energy particle are approximately the same for various liquid and solid media if the densities of these media and the charges of the elements constituting them are close to each other. For example, according to Askarijan et al. [194], the parameters of the cascade from a neutrino with energy of the order of magnitude of 1015 eV in water are L = µ −1 = 4 m and a = 2 cm (a is the radius). In this case according to the above expression, the estimate of the effective sound pressure (in Pa) in the near wave field (f = 30 kHz) is peff ≈ 0.1
E 1 ; E0 = 1016 eV . E0 r
This formula is analogous to the expression for estimation of the level of sound signal in the near wave field of the thermoradiation source produced by a cascading particle given by Berezinskii and Zatsepin [24]. If we take for example Antarctic ice as the medium where a cascade forms due to absorption of a high-energy neutrino in it, the parameters of the cascade are approximately the same as in water. If we take the numerical values cl = 4⋅103 m/s, n = 2, α = 5⋅10-5 K−1, and cε = 2⋅103 J/(kg⋅K) that correspond to the ambient temperature t = −20°C, the estimate of the effective sound pressure in the near wave field according to the above expression for σRR is given by the following expression (f = 90 kHz): eff σ RR ≈
E 1 . E0 r
One can see that the effective sound pressure in ice is approximately one order of magnitude higher than that in water, other conditions being equal. This difference is caused by the fact that the Gruneisen parameter (Γ = αcl2/cε) for ice is approximately one order of magnitude larger than that for water.
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In order to estimate the number of hydrophones necessary for construction of an acoustic neutrino detector, it is important to know the maximum distance from the cascade axis rmax where detection of the acoustic signal is still possible. That is why it is necessary to determine the optimal frequency of the frequency band for reception of the acoustic signal generated by a nuclear-electromagnetic shower. Volovik et al. [56] conducted calculations for different values of energy and substances including water. The analysis of the calculation results taking into account the number of possible events, i.e., emergence of super-high-energy particles in the detector volume and the rate of the number of events per year, have led the authors to the conclusion that the transition from water to other condensed media (detector working media) does not give essential advantages to big installations designed for detection of super-high-energy neutrinos.
7. NEUTRINO FOR GEOACOUSTICS – THE GENIUS PROJECT As construction of more and more powerful accelerators opens new opportunities for further studies of particle physics at larger and larger energies, “old” accelerators designed initially for purely basic research are applied in various fields of science and technology like medicine, biology, etc. One of the newest fields, where the advances of high-energy physics can be used effectively for both basic research and industry, may be an example of future applications of accelerators. We are talking about geophysical applications of high-energy neutrino beams. Figuratively speaking, neutrino may play the same role in this field as that of X-rays in medicine and nondestructive testing. A general scheme of geophysical application of neutrino beams is as follows [184, 248, 249]. A neutrino beam formed by an accelerator is aimed in a specified direction and travels over a considerable distance in the Earth. As it propagates, the beam generates secondary radiation of different types: muon, radio, and acoustic. Neutrinos themselves can be a probing device: neutrino absorption along its path can be a measure of the quantity of matter along it. In this case secondary emissions serve for neutrino beam detection and measurement of its parameters. In other cases (one of them we will discuss below) a neutrino beam can be used only as a source of secondary radiation, which in the course of propagating “selects” information on the characteristics of the Earth. This is the situation if the problem is to investigate surface layers of the Earth’s crust (ocean), for finding minerals for example.
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The opportunities provided by geoacoustic applications of neutrinos, i.e., use of the acoustic radiation produced by a powerful neutrino beam propagating in the Earth, for acoustic sounding of the Earth’s bowels are described briefly in this section. The realization of geoacoustic (as well as geophysical in general) applications of neutrino beams demands contradictory conditions to be satisfied: neutrino penetrating capability must be combined with a strong interaction with matter, as the neutrino beam must perform the role of a sufficiently powerful thermoradiation sound source. As calculations showed [184, 249], these conditions can be satisfied for the neutrino beams of energy Eν ≈ 1 TeV and higher. A neutrino beam of such energy produced by an accelerator is narrow and sharply directed, and this provides the high volume density of the energy released due to its interaction with matter. This creates the conditions of efficient sound generation in ground or water. The minimum energies of accelerators needed to obtain such neutrino beams depend on the accelerator type and lie within the interval of several tens of teraelectronvolts, i.e., just within the energy range of new-generation accelerators being designed now. As for the optimal energy of neutrinos for geoacoustic research, it is certainly essentially higher and may possibly be attained only in the far future. Thus, when we speak about neutrinos for geoacoustics, we speak certainly about ideas that as yet are only theoretical and at present we can speak only about projects and forecasts for accelerators of future generations. Moreover, the phenomena taking place at such an energy can be observed now only in experiments with cosmic rays (they were considered in the preceding section) and to a large extent, they need further investigation. Neutrino is a neutral particle. It is impossible to accelerate a neutrino to the necessary energy (Eν = 1 ÷ 10 TeV) directly in an accelerator, since neutrinos do not interact with an electromagnetic field. Therefore, the production of neutrinos by “elementary reactors” (unstable particles) moving at a very high velocity is used. Such a fast unstable particle can be obtained in two ways: it is possible to obtain a “slow” unstable particle, a pion (π-meson) for example, and then accelerate it before it has time to decay, or to accelerate a proton, which produces a fast unstable particle immediately after its collision with a target. Both cases are discussed in the literature devoted to neutrinos for geophysics [184]. In existing cyclic proton accelerators and accelerators, which are being designed now, proton acceleration occurs in a circular vacuum chamber, where 1013 – 1015 protons during the time ≤ 103 s are accelerated up to the maximum energy. In order to obtain a neutrino beam, these protons are brought out of the circular chamber and directed at a special target. When protons interact with the nucleus of the target, several secondary particles
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emitted under very small angles to the direction of proton motion, mainly ± ± pions (π -mesons) and kaons (k -mesons), are formed. After focusing by a system of magnetic lenses (separation of particles with charges of different sign occurs simultaneously), mesons go into a long straight vacuum + channel, i.e., a so-called decay channel, where a part of them decays: π → + + + µ + νµ and k → µ + νk . Let us take pions. Their lifetime is comparatively small (τ ≈ 2.4⋅10−8 s). However, the distance, at which they decay, is rather large because of the Lorentzian retarding of time and equals approximately lπ = 56 (Eπ/TeV) km. For example, only about 20% of pions decay at energy Eπ = 1 TeV at the length lπ = 1 km. In the process of pion decay, a muon is born apart from a neutrino, and in the process of muon decay, neutrinos and antineutrinos are born in their turn. However, muons make almost no contribution to a neutrino flux, while their lifetime is approximately two orders of magnitude longer than the lifetime of pions. As the result of a chain of transformations p → π → ν, the average energy of a neutrino in a beam is approximately 20 times smaller than the energy of primary protons. Thus, although it is suggested to use first of all cyclic accelerators to attain the maximum energy of accelerated particles, these accelerators have considerable disadvantages as the sources of neutrino beams – degradation of energy of primary particles (protons), a large length of the decay channel, etc. Unique opportunities may be provided by linear accelerators, where particle acceleration occurs during the time of a single flight of a particle. The calculations performed in P. N. Lebedev Physical Institute, Russian Academy of Sciences, [184] showed that pion accelerators could be very effective sources of neutrino beams for geophysical research. Pion accelerators are at least four times more effective than proton accelerators in the ratio of the initial energy of protons and the energy of unstable particles, i.e., pions. In contrast to annular accelerators, a linear accelerator can operate at high rates of pulse repetition (70–100 Hz) that provides an opportunity to obtain a high average intensity of the neutrino beam. As we know now, this is important from the point of view of efficiency of radiation-acoustic conversion. An important advantage of linear pion accelerators in comparison with proton accelerators is the smaller energy of the primary beam that makes its deflection easier. Finally, one can use “packages” of particles with a very high density in a linear accelerator, which provides the opportunity to obtain short pulses with a number of particles sufficient for detection of a radiation-acoustic signal. Speaking about neutrino beams we have always meant their pion “origin”. Meanwhile, as we have noted earlier, kaons are also the source of neutrinos in the process of interaction of accelerated protons with the nuclei of the target. However, estimations show that the role of kaon neutrinos is
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rather insignificant. Their contribution may be noticeable only far from the axis of a neutrino beam [184]. It has been noted also that the necessity of a large-length decay channel (1 ÷ 10 km) is a big disadvantage of a superpowerful proton accelerators serving as a source of a neutrino beam. However, as has been demonstrated by physicists from Lebedev Physical Institute, there may be no need of a decay channel if one uses so-called “direct” neutrinos born in the process of decay of charmed particles (Λ, D, etc.) discovered rather recently. This is still more evidence of the fact that the results of basic research in the field of high-energy physics can change our views on the sources of neutrino beams. Rujula et al., who introduced the idea of using a thermoacoustic signal generated in rocks by a neutrino beam for geological research, were apparently the first who made estimates of proton accelerator parameters needed for the creation of the required neutrino beam and the acoustic signal generated by it [249]. On these estimates they based their proposal for the GENIUS (Geological Exploration by Neutrino Induced Underground Sound) project. They considered the possibility of construction of a circular accelerator for proton energy Ep = 3 ÷ 20 TeV, which was named Geotron in contrast to the proton accelerator for the energy Ep = 1 TeV named Tevatron, since in the latter case a new generation accelerators for teraelectronvolt energy are meant. Thus, the GENIUS project can be the second example of an immense project in the field of high-energy physics and radiation acoustics. We may consider the DUMAND project to be the first. While it is significant that a proton accelerator for energy Ep = 1 TeV will every few minutes “eject” 1014 protons with total kinetic energy 1 MJ, then these parameters are even more impressive for the Geotron. For example, the number of protons per pulse is 1015 and the total energy is ~109 J. If the most modern superconductors are used for the construction of the accelerator magnet system, the radius of its circular vacuum channel will be equal to approximately 6 km for the proton energy Ep = 10 TeV, and for the proton energy Ep = 20 TeV it will be 12 km, respectively. The length of the decay channel is estimated as l = 7.5 km (Ep/10 TeV), i.e., about l = 7.5 km at Ep = 10 TeV and l = 15 km at Ep = 20 TeV. The synchrotron radiation power is a very important parameter of the accelerator. Rujula et al. [249] estimate this power according to the formula Ws = 6⋅10−14 NEp4/R2 ; here N is the number of particles (protons), R is the radius of the circular channel (in kilometers), and Ep is the proton energy (in TeV). Thus, at Ep = 10 TeV, N = 1015, and R = 6 km, we have Ws = 16 kW and at Ep = 20 TeV, Ws = 64 kW, correspondingly. In order to protect people nearby from synchrotron radiation, and taking into account the fact that the problem of controlling a decay channel of length of about 15 km is not the “simplest” one, Rujula et al. [249] consider as one of the
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possibilities the construction of a circular chamber and decay channel of a proton accelerator for energy Ep = 10 ÷ 20 TeV in the sea. A scheme of a neutrino geoacoustical experiment is given in Fig. 11.16. A neutrino beam propagating deep in the Earth is the source of acoustic waves and sound detection has to be performed by a geophone array at the Earth’s surface or a hydrophone array if measurements are conducted at sea. As for the parameters of a neutrino beam, since only several percent of pions have time to decay in the decay channel, according to estimations [249] the beam should contain only 1013 neutrinos if there are 1015 particles in the proton beam, and the neutrino energy should be approximately Eν = 0.3 TeV at Ep = 10 TeV. The next formula is given [249] for the radius a of the neutrino beam: a = 10.8 m (L/1000 km)⋅(10 TeV/Ep). Here L is the distance to the accelerator (in km). If L = 1000 km, then a = 10 m. This means that the neutrino flux in the beam at this distance constitutes ~1010 neutrinos per 1 m2.
Figure 11.16 Scheme of neutrino-geoacoustical sounding [249]. (1) A neutrino beam, (2) the Earth surface, (3) a geophone (hydrophone) array; R is the distance to the geophone array and d is the diameter of the neutrino beam.
The calculation of acoustic radiation from a neutrino beam, which was conducted by Rujula et al. [249], was based on the equation of thermoradiation sound generation in the form 2 2 2 c ∆ − ∂ p (r , t ) = − αc 1 (1 + σ ) ∂ε , C p 3 (1 − σ ) ∂t ∂t 2
where α is the coefficient of volumetric thermal expansion of a medium, Cp is the specific heat capacity, σ is Poisson’s ratio, c is the sound velocity, and ε is the cubic density of energy dissipated in a medium. We should note that the factor (1 + σ)/[3(1 − σ)] is approximately equal to one and the quantity αc2/Cp = Γ is the Gruneisen parameter of a medium. Estimates show that about 100 interactions with the medium substance per 1 cm occur at proton energy Ep = 10 TeV in a neutrino beam. This means that, finally, the ionization losses constitute ~ 100 erg/cm. At distance L = 1000 km from the Geotron, the cubic density of the energy
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released in a medium is approximately equal to ε = 3⋅10−5 erg/cm3. In many cases the Gruneisen parameter is about one and the estimate for the amplitude of the acoustic pulse is p ≈ 3⋅10−5 dyne/cm2. The acoustic signal has the characteristic form of the so-called N-wave. Rujula et al. [249] give also the following estimate for the N-wave amplitude: for Ep = 10 TeV, L = 1000 km, R = 1 km (R is the depth of the neutrino beam in Earth, the distance to the geophone) and rock salt NaCl (for example) p ≈ 1.8⋅10−5 dyne/cm2 and for the same parameters and Ep = 20 TeV a more exact calculation gives p = 4.1⋅10−5 dyne/cm2. A certain increase (more than twice) of the signal amplitude is caused in particular by the fact that as the neutrino energy grows, the beam diameter decreases, and the cubic density of the energy released in a substance increases additionally. An important characteristic of an acoustic signal is its frequency spectrum. The spectrum density of an acoustic signal has the form p(ω ) → ω 1 / 2 exp( −ω / ω 0 ) , where ω0 = c/a and a is the radius of the neutrino beam. The maximum spectral density pmax(ω) corresponds to the frequency ω/2 and 〈ω 〉 = 1.50ω 0 , 〈ω 2 〉 2 = 3.75ω 02 , ∆ω = [〈ω 2 〉 − 〈ω 〉 2 ]1 / 2 = 1.22ω 0 . The characteristic frequency (in Hz) is given by an expression f0 =
ω0 c . = 2π 2πa
Rujula et al. [249] give the next expression for the radius of the neutrino beam a: L 10 TeV a = 10.8 m . 1000 km E p It follows from the formulae given above that
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L Ep c 〈 f 〉 = 23 , 1 km/s 1000 km 10 TeV where 〈f〉 is expressed in hertz. For L = 1000 km, Ep = 10 TeV, and c = 4.74 km/s (rock salt) we can obtain 〈f〉 = 109 Hz and the frequency band ∆f = 89 Hz. These estimates (and the estimate of the acoustic signal amplitude especially) are very rough as has been stressed by Rujula et al. [249].
Figure 11.17 Scheme of neutrino-geoacoustical mineral prospecting [249]. (1) The Earth (oceanic) surface, (2) horizontal sedimentary, (3) impervious sedimentary, (4) gas and oil, (5) water, (6) a mixture of water, oil, and gas, (7) a rock base rich with organic substances, and (8) the propagation direction of the sound waves excited by a neutrino thermoradiation source of sound.
The signal amplitude is very small. However, the fundamental quantity from the point of view of signal reception is the signal-to-noise ratio. The level of noise (seismic noise or oceanic noise if we mean underwater reception) within the range of the characteristic frequencies (~100 Hz) can be approximately five orders of magnitude higher than the signal level, but nevertheless, signal reception under these conditions may be not at all hopeless. The noise stability may be increased using a lattice (an array) with a large number of receivers. If the noise at the array receivers is uncorrelated while the signal at all receivers of the array is correlated, then the noise 2 stability of reception increases proportionally to n1/ , where n is the number
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of receivers in the array. Furthermore, one can use coherent signal integration to increase noise stability. As the positions of the sound source and receiving array are fixed, one can take a big enough integration time. In particular, utilization of a linear pion accelerator in the Geotron design would contribute to this. It would provide an opportunity to increase the repetition rate of neutrino (acoustic) pulses. In real conditions the amplitude of the acoustic signal generated by a neutrino beam can be several orders of magnitude higher. As was noted by Tsarev and Chechin [184] and Balitskii et al. [15], a natural solid medium can behave as an active medium: the initial acoustic radiation by a neutrino beam can act as a kind of a trigger mechanism initiating the rise of acoustic emission in a medium. The signal from this emission is essentially higher than the signal generated by a neutrino beam and the “useful” signal is the signal of acoustic emission. Figure 11.17 presents one of the possible schemes of neutrino geoacoustical mineral prospecting. A neutrino beam is transmitted through various geological strata with different values of Gruneisen coefficient. Measuring the change of the acoustic signal by a lattice of geophones while moving it along the beam track, one can obtain information on the type of rocks and minerals.
Conclusion We can state that theoretical and experimental studies of thermoradiation mechanism of sound generation by penetrating radiation in a condensed medium have advanced quite far by now. The processes of sound generation in the case of continuous (modulated) and pulsed action of penetrating radiation on a substance are studied. Basic laws of formation of acoustic signals are established and the connection of the characteristics of these signals with the parameters of radiation and thermodynamic, radiation, and acoustic properties of substances are revealed. The optimal conditions and efficiency of thermoradiation generation of sound are determined. The particular features of sound generation by a particle beam moving along the surface of a liquid or solid in the cases of subsonic, transonic, and supersonic velocities of beam motion and the arbitrary form of modulation of radiation intensity in the beam are studied. The possibility of production of sound sources operating in a broad frequency range from sonic to hypersonic frequencies in liquids or solids is demonstrated. It is possible to change the frequency, directivity, and intensity (power) of a radiationacoustic sound source by selecting the type of radiation and controlling the parameters of a radiation beam. The processes of sound generation by single high-energy particles in a substance are studied. The theory of thermoradiation sound generation in condensed media has been confirmed reliably by experiments. This provides grounds for justified selection of sources of penetrating radiation for the solution of practical problems of thermoradiation sound generation such as radiation-acoustic microscopy and visualization, radiation-acoustic dosimetry, application of radiation-acoustic effects to nondestructive testing, etc. 339
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There are the prospects of carrying out immense and maybe fantastic (as it may seem now) projects, i.e., the DUMAND project in neutrino astrophysics and the GENIUS project in neutrino geoacoustics. Radiation excitation of sound can be useful also in other situations, which may seem unusual, e.g., for excitation of sound pulses in cosmic bodies by laser radiation from the Earth or powerful beams of penetrating radiation from space platforms and creation of vertical underwater acoustic arrays in the ocean, with efficient height or length of about several tens of meters (if the sources of laser radiation operating in the blue-green optical range are used) and of thousands of meters (beams of muons and neutrinos). These problems do not have “simple” technological solutions if traditional radiators are used. Summarizing, we have to note however that only linear theory has been treated in this book. As one can see, this theory describes radiation-acoustic effects when the intensity of penetrating radiation is small. And the phenomenon of interaction of penetrating radiation with a substance is nonlinear by its nature. From this point of view both the theory and the conditions of experiments (practical application of the conclusions of the linear theory) are restricted within the framework of the perturbation method. Furthermore, only comparatively simple “model” problems have been considered within the framework of this linear theory. As for thermoradiation generation of sound in solids, only the model of an isotropic solid has been considered. However, in practice one has to deal with solids of complex structure like semiconductors, piezoelectrics, ferroelectrics, ferromagnetics, etc., where it is important to take into account the interaction of penetrating radiation with various subsystems, i.e., the lattice and electron-hole, spin, etc., subsystems, as well as the interaction of these subsystems in complex solid structures. It is possible that this research will lead into new fields of science such as radiation acousto-electronics and radiation magneto-acoustics. Broad prospects for development of new technology may be opened by the studies of nonlinear radiation-acoustic effects, which arise in a substance when there are no changes of the aggregate state of this substance and no phase transitions, but the expansion rate of the volume of the substance absorbing radiation is large enough or change of its thermophysical properties occurs. The heated region of a medium “works” in this case as a source of finite amplitude waves, which can transform in their turn into shock waves. Mechanical, physical, and chemical properties can change under the impact of these shock waves that may be used for new technologies. If a powerful, penetrating radiation affects a substance in the conditions when phase transitions occur and substance evaporation or optical breakdown (under the action of powerful laser radiation) take place, shock and sound waves of huge amplitude with parameters unattainable by
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traditional means can rise. Radiation sources of great power are already developed or being developed now (regretfully they are intended for the use in ray and beam weapons as has been reported [87]). Hopefully, these sources will never be used for their “direct” purpose but applied to peaceful tasks including radiation-acoustic technology.
REFERENCES
1.
S. M. Avanesyan et al. “Generation of surface acoustic waves by deformation and thermal mechanisms in the case of optical action on silicon”, Akusticheskii Zhurnal, 1986, v. 32, No.4, pp. 562 – 564. 2. Yu. M. Ado “Accelerators of charged particles of high energy”, Uspekhi Fizicheskikh Nauk, 1985, v. 145, No. 1, pp. 87 – 112. 3. Yu. M. Annenkov, B. F. Stolyarenko, and T. S. Fransul’yan “Detection of destruction of radiation defects by mechanical stress pulses generated in crystals by a powerful electron beam of nanosecond length”, Zhurnal Tekhnicheskoi Fiziki, 1986, v. 56, issue 6, p. 1208. 4. A. I. Anoshin “On acoustic radiation of ionizing particles in water”, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1977, v. 47, No. 10, pp. 2186 – 2192. 5. V. V. Apollonov et al. “Thermoelastic action of pulsed-periodic laser radiation upon the surface of a solid”, Kvantovaya Elektronika, 1982, v. 9, No. 2, pp. 343 – 353. 6. G. A. Askar’yan “Hydrodynamic radiation from tracks of ionizing particles in stable liquids”, Atomnaya Energiya, 1957, v. 3, No. 8, pp. 152 – 153. 7. G. A. Askar’yan “Radiation of surface and bulk compression waves in the process of collision of a non-relativistic electron flux with a surface of a dense medium”, Zhurnal Tekhnicheskoi Fiziki, 1959, v. 29, pp. 267 – 269. 8. G. A. Askar’yan and B. A. Dolgoshein “Acoustic detection of high-energy neutrino at large depth”, Preprint No. 160, Lebedev Physical Institute, USSR Academy of Sciences, Moscow, 1976. 9. G. A. Askar’yan and B. A. Dolgoshein “Acoustic detection of high-energy neutrino”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1977, v. 25, No. 5, pp. 617 – 620. 10. G. A. Askar’yan and B. A. Dolgoshein “Microelectrostriction in the process of medium ionization”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1978, v. 28, No. 10, pp. 617 – 620. 343
344
REFERENCES
11. G. A. Askar’yan et al. “Acoustic detection of high-energy neutrino”, Preprint No. 140, Lebedev Physical Institute, USSR Academy of Sciences, Moscow, 1976. 12. “Astrophysics of cosmic rays”, Ed. by V. L. Ginzburg, Nauka, Moscow, 1984. 13. V. L. Auslender et al. “Powerful electron accelerators and radiation technology”, Vestnik AN SSSR, 1981, No. 6, pp. 48 – 57. 14. A. I. Akhiezer and M. P. Rekalo “Elementary particles”, Nauka, Moscow. 15. V. A. Balitskii et al. “Sound excitation by an electron beam in natural solid media”, Preprint No. 138, Lebedev Physical Institute, USSR Academy of Sciences, Moscow, 1988. 16. V. A. Balitskii et al. “Sound generation by an electron beam in water”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 694 – 695. 17. Baranov et al. “On acoustic radiation initiated in a liquid by fission fragments”, Atomnaya Energiya, 1982, v. 52, No. 5, pp. 335 – 336. 18. F. G. Bass and I. M. Fuks “Wave scattering at a statistically uneven surface”, Nauka, Moscow, 1972. 19. S. Z. Belen’kii “Avalanche processes in cosmic rays”, GTTI, Moscow, 1948. 20. D. Bell and S. Preston “Theory of nuclear reactors”, Atomizdat, Moscow, 1974. 21. H. Batteman and A. Erdelyi “Tables of integral transforms”, McGraw-Hill, New York – Toronto – London, 1954. 22. V. S. Berezinskii et al. “Astrophysics of cosmic rays”, Nauka, Moscow, 1984. 23. V. S. Berezinskii and A. Z. Gazizov “Cosmic neutrino and possibility of search − for W -boson with the mass 30 – 100 GeV in underwater experiments”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1977, v. 25, No. 5, pp. 276 – 278. 24. V. S. Berezinskii and G. T. Zatsepin “Possibility of experiments with cosmic neutrinos of very high energy. The DUMAND project.”, Uspekhi Fizicheskikh Nauk, 1977, v. 122, No. 1 (500), p. 3. 25. A. A. Bespal’ko and G. I. Gering “Acoustic dosimetry of intense electron beams”, Zhurnal Tekhnicheskoi Fiziki, 1980, v. 50, No. 1, pp. 213 – 215. 26. A. A. Bespal’ko “Acoustic radiation of solids under the action of intense electron beams of nanosecond length”, Dr. Sc. Thesis, Sverdlovsk UPI, 1983. 27. A. A. Bespal’ko and G. I. Gering “Acoustic dosimetry of electron beams”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1980, v. 25, No. 2, pp. 115 – 117. 28. S. V. Blazhevich et al. “Investigation of acoustic effect of interaction of relativistic electrons with thin targets”, Fizika Tverdogo Tela, 1975, v. 17, No. 12., pp. 3636 – 3638. 29. A. I. Bozhkov and F. V. Bunkin “Sound generation in a liquid in the process of absorption of intensity-modulated laser radiation in it”, Kvantovaya Elektronika, 1975, v. 2, No. 8, pp. 1763 – 1776. 30. A. I. Bozhkov, F. V. Bunkin, and L. L. Gyrdev “Effect of waves at the surface of a liquid on the sound field excited in it by intensity-modulated laser radiation”, Kvantovaya Elektronika, 1976, v. 3, No. 7, pp. 1494 – 1500. 31. A. I. Bozhkov, F. V. Bunkin, and Al. A. Kolomenskii “Study of sound field of supersonic optoacoustic array”, Kvantovaya Elektronika, v. 4, No. 6, pp. 942 – 943.
REFERENCES
345
32. A. I. Bozhkov, F. V. Bunkin, and Al. A. Kolomenskii “Sound radiation at finite tracks in the case of optoacoustic effect”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1978, v. 4, No. 21, pp. 1283 – 1286. 33. A. I. Bozhkov et al. “Laser excitation of powerful sound in a liquid”, Studies in Hydrophysics, Proceedings of Lebedev Physical Institute, 1984, v. 156, pp. 123 – 176. 34. A. I. Bozhkov and Al. A. Kolomenskii “Sound field of an optoacoustic array moving with sub- or supersonic velocity”, Kvantovaya Elektronika, 1978, v. 5, No. 12, pp. 2577 – 2586. 35. A. I. Bozhkov, V. G. Mikhalevich, and A. I. Malyarovskii “Investigation of the wave zone of a thermooptical radiator of sound in a liquid”, Akusticheskii Zhurnal, 1979, v. 25, No. 6, pp. 820 – 824. 36. I. A. Borshkovskii and V. D. Volovik “Study of excitation of acoustic waves in metals by electrons and protons”, Izvestiya Vuzov, Fizika, 1973, No. 10, pp. 72 – 76. 37. I. A. Borshkovskii et al. “Investigation of acoustic waves in metals by fast charged particles and γ-quanta”, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1972, v. 63, No. 4 (10), pp. 1338 – 1342. 38. I. A. Borshkovskii, V. D. Volovik, and V. T. Lazurik-El’futsin “Acoustic peak of protons in a solid plate”, PMTF, 1975, No. 2, pp. 138 – 140. 39. I. A. Borshkovskii et al. “Excitation of ultrasonic waves in the process of passage of fast electrons through metals”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1971, v. 13, No. 10, pp. 546 – 549. 40. L. M. Brekhovskikh “Waves in layered media”, Publishing House of the USSR Academy of Sciences, Moscow, 1957. 41. F. V. Bunkin and V. M. Komissarov “Optical excitation of sound waves”, Akusticheskii Zhurnal, 1973, v. 19, No. 3, pp. 305 – 320. 42. F. V. Bunkin, V. G. Mikhalevich, and G. P. Shipulo “Generation of monochromatic sound in water in the process of absorption of laser radiation in it”, Kvantovaya Elektronika, 1976, v. 3, No. 2, pp. 441 – 443. 43. F. V. Bunkin and M. P. Tribel’skii “Non-resonance interaction of powerful optical radiation with a liquid”, Uspekhi Fizicheskikh Nauk, 1980, v. 130, No. 2, pp. 193 – 239. 44. L. V. Burmistrova et al. “Transfer function technique in problems of thermooptical sound excitation”, Akusticheskii Zhurnal, 1978, v. 24, No. 5, pp. 655 – 663. 45. D. I. Vaisburd et al. “High-energy electronics of solids”, Nauka, Novosibirsk, 1982. 46. “Introduction to radiation acoustics”, Publishing House of Khar’kov University, Khar’kov, 1986. 47. V. I. Veksler “A new method of acceleration of relativistic particles”, Doklady Akademii Nauk SSSR, 1944, v. 43, No. 8, pp. 346 – 348. 48. E. P. Velikhov et al. “Amplification of a surface elastic wave accompanied by laser radiation in a solid”, Preprint of the Kurchatov Institute of Atomic Energy, 1983, No. 15.
346
REFERENCES
49. E. P. Velikhov et al. “Amplification of a surface elastic wave accompanied by laser radiation in a solid”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1983, v. 38, No. 10, pp. 483 – 486. 50. I. A. Veselovskii et al. “Effect of phase transitions on photoacoustic effect in the process of action of laser radiation on condensed media”, Kvantovaya Elektronika, 1985, v. 12, No. 2, pp. 382 – 383. 51. V. S. Vladimirov “Equations of mathematical physics”, Nauka, Moscow, 1967. 52. V. D. Volovik “Excitation of elastic waves by neutrons in substances unstable with respect to fission”, Izvestiya Vuzov, Fizika, 1975, No. 7, pp. 145 – 147. 53. V. D. Volovik and V. N. Ivanov “On the problem of thermoelastic dosimetry of beams of charged particles”, Zhurnal Tekhnicheskoi Fiziki, 1975, v. 45, No. 8, pp. 1789 – 1791. 54. V. D. Volovik et al. “Thermoelastic effect of a fast particle in a solid”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1974, v. 19, pp. 135 – 138. 55. V. D. Volovik, A. I. Kalinichenko, and V. T. Lazurik-El’futsin “On the nature of the acoustic effect of fast charged particles in metals”, Problemy Yadernoi Fiziki i Kosmicheskikh Luchei, 1976, v. 4, pp. 80 – 86. 56. V. D. Volovik et al. “Acoustic calorimetry of super-high-energy particles”, Izvestiya AN SSSR, physical series, 1980, v. 44, No. 3, pp. 586 – 589. 57. V. D. Volovik et al. “On generation of elastic waves by beams of charged particles in stable liquids”, Zhurnal Tekhnicheskoi Fiziki, 1979, v. 49, No. 6, pp. 1343 – 1345. 58. V. D. Volovik et al. “Thermoacoustic effect of a cascading particle in water”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1978, v. 4, No. 10, pp. 611 – 615. 59. V. D. Volovik and V. I. Kobizskoi “Acoustic effect of pulsed beams of charged particles in halogen-containing stable liquids”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1976, v. 2, No. 2, pp. 66 – 69. 60. V. D. Volovik and V. I. Kobizskoi “Acoustic effect in the process of pulsed radiolysis of liquids”, Zhurnal Tekhnicheskoi Fiziki, 1977, v. 47, No. 12, pp. 2555 – 2560. 61. V. D. Volovik and V. I. Kobizskoi “Nonlinear radiation-acoustic phenomena in the process of pulsed radiolysis”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1979, v. 5, No. 16, pp. 995 – 997. 62. V. D. Volovik, V. I. Kobizskoi, and G. F. Popov “Acoustic effect from superhigh-energy particles in some stable liquids”, Izvestiya AN SSSR, Physical series, 1976, v. 40, No. 5, pp. 1065 – 1067. 63. V. D. Volovik and V. T. Lazurik-El’futsin “Dynamic effect in the process of passage of elementary particle beam in solids”, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1972, v. 63, No. 5 (11), pp. 1776 – 1779. 64. V. D. Volovik, V. V. Petrenko, and G. F. Popov “On the mechanism of hydrodynamic radiation of charged particles in water”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1977, v. 3, No. 10, pp. 459 – 462. 65. V. D. Volovik, V. V. Petrenko, and G. F. Popov “On the nature of elastic waves from fluxes of charged particles in liquids and on acoustic detection of high-energy particles”, Preprint No. P0092, Institute of Nuclear Research of the USSR Academy of Sciences, Moscow, 1978.
REFERENCES
347
66. V. D. Volovik and G. F. Popov “Elastic waves in stable liquids from pulsed beams of charged particles”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1976, v. 1, No. 13, pp. 601 – 603. 67. P. I. Golubnichii, G. S. Kalyuzhnyi, and V. I. Yakovlev “On the possibility of simulation of acoustic effects arising in the process of development of electron cascades in liquids with the help of lasers”, Preprint No. 90, Lebedev Physical Institute of the USSR Academy of Sciences, Moscow, 1979. 68. P. I. Golubnichii, G. S. Kalyuzhnyi, and S. D. Korchikov “On the mechanism of generation of acoustic radiation in liquids by beams of ionizing particles”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1981, v. 7, No. 5, pp. 272 – 276. 69. P. I. Golubnichii et al. “On the possibility of simulation of acoustic effects arising in the process of passage of ionizing particle beams through liquids with the help of laser radiation”, Brief Communications in Physics, Lebedev Physical Institute of the USSR Academy of Sciences, 1978, No. 8, pp. 19 – 23. 70. P. I. Golubnichii et al. “Measurement of coefficients of sound generation at an electron beam in solids”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 700 – 701. 71. P. I. Golubnichii, V. G. Kublenko, and V. I. Yakovlev “On the mechanism of sound generation in liquids by fission fragments”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 703 – 704. 72. V. I. Gol’danskii, E. Ya. Lintsburg, and N. A. Yampol’skii “On the hydrodynamic effect in the process of passage of fission fragments through condensed matter”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1976, v. 21, No. 6, pp. 365 – 367. 73. Yu. A. Gol’fand and E. P. Likhtman, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1971, v. 13, pp. 452 – 454. 74. D. O. Gorelik and B. B. Sakharov “Optoacoustic effect in physico-chemical measurements”, Publishing House of the State Committee for Standards, Moscow, 1969. 75. V. S. Gorodetskii et al. “On sound generation by laser pulses”, Kvantovaya Elektronika, 1978, v. 5, No. 11, pp. 2396 – 2401. 76. A. P. Grinberg “To the theory of invention and design of accelerators (1922 – 1932)”, Uspekhi Fizicheskikh Nauk, 1975, v. 117, pp. 333 – 362. 77. V. T. Grinchenko and V. V. Meleshko “Harmonic vibrations and waves in elastic bodies”, Naukova Dumka, Kiev, 1981. 78. V. G. Grishin “Quarks and hadrons in interaction of high-energy particles”, Energoatomizdat, Moscow, 1988. 79. V. E. Gusev and A. A. Karabutov “Theory of excitation of Rayleigh waves due to absorption of optical pulses in semiconductors”, Fizika i Tekhnika Poluprovodnikov, 1986, v. 20, No. 6, pp. 1070 – 1075. 80. M. I. Guseva and Yu. V. Martynenko “Radiation blistering”, Uspekhi Fizicheskikh Nauk, 1981, v. 135, No. 4, pp. 671 – 691. 81. A. A. Davydov and A. I. Kalinichenko “To the inverse problem of radiation acoustics”, Akusticheskii Zhurnal, v. 31, No. 5, pp. 704 – 705. 82. A. A. Davydov, A. I. Kalinichenko, and V. T. Lazurik “On radiation-acoustic effects in media with ellipsoidal inclusions”, Problems of nuclear physics and cosmic rays, 1984, v. 21, pp. 43 – 49.
348
REFERENCES
83. A. A. Davydov and V. T. Lazurik “Sound excitation in a heterogeneous material by penetrating radiation pulses”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 705 – 706. 84. I. A. Danil’chenko et al. “Investigation of acoustic signal characteristics in the near wave field of a proton beam”, in: Radiation Acoustics, Nauka, Moscow, 1987, pp. 51 – 57. 85. I. A. Danil’chenko et al. “Investigation of acoustic signal characteristics in the near wave field of a proton beam in water”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 706 – 707. 86. E. V. Darinskaya et al. “Study of dynamics of dislocations in the process of crystal deformation by super-short radiation pulses in an electron beam”, Fizika Tverdogo Tela, 1982, v. 24, No. 3, pp. 940 – 941. 87. “Report to American Physical Society of the expert group on scientific and technological aspects of beam weapons”, Uspekhi Fizicheskikh Nauk, 1988, v. 155, No. 4, pp. 559 – 679. 88. V. P. Dokuchaev “Radiation of sound waves by a body moving along a circle and a rotating vane of a simple shape”, Akusticheskii Zhurnal, 1965, v. 11, No. 3, pp. 324 – 333. 89. I. V. Dorman “Cosmic rays, accelerators, and new particles”, Nauka, Moscow, 1988. 90. T. A. Dunina et al. “On the near field of a pulsed thermoacoustic array”, Akusticheskii Zhurnal, 1979, v. 25, No. 1, pp. 60 – 64. 91. T. A. Dunina et al. “Investigation of thermooptical generation of sound by nanosecond laser pulses”, Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1979, v. 5, No. 16, pp. 986 – 989. 92. A. M. Dykhne and B. P. Rysev “On possibility of excitation of elastic surface waves of large amplitude in solids under the thermal impact of laser radiation”, Surface. Physics, Chemistry, Mechanics. 1983, No. 6, pp. 17 – 21. 93. S. V. Egerev et al. “Sound generation by long laser pulses”, Akusticheskii Zhurnal, 1979, v. 25, No. 2, pp. 220 – 226. 94. S. V. Egerev and K. A. Naugol’nykh “On acoustooptic phenomena in a liquid with gas bubbles”, Akusticheskii Zhurnal, 1977, v. 23, No. 5, pp. 738 – 742. 95. I. B. Esipov “Sound radiation by a thermal source moving at a supersonic velocity”, Akusticheskii Zhurnal, 1977, v. 23, No. 1, pp. 164 – 165. 96. V. P. Zharov and V. S. Letokhov “Laser optoacoustic spektroscopy”, Nauka, Moscow, 1984. 97. I. I. Zalyubovskii, A. I. Kalinichenko, and V. T. Lazurik “Introduction to radiation acoustics”, Vishcha Shkola, Kiev, 1986. 98. Ya. B. Zel’dovich “Classification of elementary particles and quarks intended for ‘pedestrians’”, Uspekhi Fizicheskikh Nauk, 1965, v. 86, pp. 303 – 314. 99. V. E. Zuev “Propagation of visible and infrared waves in atmosphere”, Sovetskoe Radio, Moscow, 1970. 100. M. A. Isakovich “General acoustics”, Nauka, Moscow, 1973. 101. I. M. Kaganov, I. M. Lifshits, and L. V. Tanatarov “Relaxation between electrons and lattice”, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1956, v. 37, issue 2 (8), pp. 232 – 237.
REFERENCES
349
102. I. M. Kaganova and M. P. Kaganov “Cherenkov radiation of sound by a particle moving through metals”, Fizika Tverdogo Tela, 1973, v. 15, No. 7, pp. 2119 – 2125. 103. A. I. Kalinichenko and V. T. Lazurik “Thermoacoustic effect of radiation beams in heterogeneous media”, Zhurnal Tekhnicheskoi Fiziki, 1981, v. 51, No. 11, pp. 2449 – 2450. 104. A. I. Kalinichenko and V. T. Lazurik “Propagation of acoustic pulses produced by fission fragments and fast neutrons in matter”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 708 – 709. 105. A. A. Karabutov “Laser excitation of surface waves: a new field in optoacoustic spectroscopy of solids”, Uspekhi Fizicheskikh Nauk, 1985, v. 147, No. 3, pp. 605 – 620. 106. A. A. Karabutov, O. V. Rudenko, and E. B. Cherepetskaya “To the theory of thermooptical generation of non-stationary acoustic fields”, Akusticheskii Zhurnal, 1979, v. 25, No. 3, pp. 383 – 394. 107. S. G. Kasoev et al. “Sound generation by laser radiation in a liquid half-space with two types of boundary unevenness”, Akusticheskii Zhurnal, 1979, v. 25, No. 3, pp. 401 – 407. 108. S. G. Kasoev and L. M. Lyamshev “Sound generation due to absorption of modulated laser radiation in a liquid half-space with large-scale boundary unevenness”, 1977, v. 23, No. 2, pp. 265 – 272. 109. S. G. Kasoev and L. M. Lyamshev “To the theory of sound generation in a liquid by laser pulses”, Akusticheskii Zhurnal, 1977, v. 23, No. 6, pp. 890 – 898. 110. S. G. Kasoev and L. M. Lyamshev “On sound generation in a liquid by laser pulses of arbitrary shape”, Akusticheskii Zhurnal, 1978, v. 24, No. 4, pp. 534 – 538. 111. S. G. Kasoev and L. M. Lyamshev “On some particular features of sound generation by laser pulses”, Proceedings of the 7th All-Union Symposium on Diffraction and Propagation of Waves, Nauka, Moscow, 1977, v. 1, p. 912. 112. Al. A. Kolomenskii “Sound radiation by an optoacoustic source moving along a finite trajectory”, Akusticheskii Zhurnal, 1979, v. 25, No. 4, pp. 547 – 555. 113. Al. A. Kolomenskii “Transient radiation of sound by an optoacoustic source”, Preprint No. 123, Lebedev Physical Institute of the USSR Academy of Sciences, Moscow, 1979. 114. V. A. Krasil’nikov and V. V. Krylov “Introduction to physical acoustics”, Nauka, Moscow, 1984. 115. V. V. Krylov and V. I. Pavlov “Thermooptical excitation of surface acoustic waves in solids”, Akusticheskii Zhurnal, 1982, v. 28, No. 6, pp. 836 – 837. 116. V. D. Kulikov and V. M. Lisitsyn “Polarization-optical detection of acoustic waves generated by strong electron beams in solids”, Zhurnal Tekhnicheskoi Fiziki, 1983, v. 53, issue 12, p. 2417. 117. V. D. Kupradze “Methods of potential in the elasticity theory”, Fizmztgiz, Moscow, 1963. 118. B. F. Kur’yanov “Sound scattering at a rough surface with two types of unevenness”, Akusticheskii Zhurnal, 1962, v. 8, No. 3, pp. 325 – 333.
350
REFERENCES
119. G. W. Kaye and T. H. Laby “Tables of physical and chemical constants”, Longmans, Green, and Co., London – New York – Toronto, 1962. 120. L. D. Landau and E. M. Lifshits “Theoretical physics. Mechanics of continuous media.”, 2nd edition, Gostekhizdat, Moscow, 1954. 121. L. D. Landau and E. M. Lifshits “Theoretical physics. Theory of elasticity.”, 4th edition, v. 7, Nauka, Moscow, 1987. 122. L. D. Landau and E. M. Lifshits “Theoretical physics. Electrodynamics of continuous media.”, 2nd edition, v. 8, Nauka, Moscow, 1982. 123. E. M. Lifshits and L. P. Pitaevskii “Physical kinematics”, Nauka, Moscow, 1979. 124. V. N. Lugovoi and V. N. Strel’tsov “Sound disturbances in a medium due to motion of a light focus”, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1973, v. 65, No. 4, pp. 1407 – 1415. 125. Yu. P. Lysanov “Sound scattering by uneven surfaces”, in: Ocean Acoustics, Ed. by L. M. Brekhovskikh, Nauka, Moscow, 1974, pp. 231 – 330. 126. V. E. Lyamov, U. Madvaliev, and R. E. Shikhlinskaya “Photoacoustic spectroscopy of solids”, Akusticheskii Zhurnal, 1979, v. 25, p. 427. 127. L. M. Lyamshev “Lasers in acoustics”, Uspekhi Fizicheskikh Nauk, 1987, v. 151, No. 3, p. 479, and Vestnik AN SSSR, 1984, No. 8, p. 97. 128. L. M. Lyamshev “Laser thermooptical excitation of sound”, Nauka, Moscow, 1989. 129. L. M. Lyamshev “Optoacoustic sources of sound”, Uspekhi Fizicheskikh Nauk, 1981, v. 135, No. 4, pp. 637 – 669. 130. L. M. Lyamshev “To the theory of optical generation of sound in liquids and solids”, Akusticheskii Zhurnal, 1977, v. 23, No. 1, pp. 169 – 170. 131. L. M. Lyamshev “To the problem of the reciprocity principle in acoustics”, Doklady AN SSSR, 1959, v. 125, No. 6, pp. 1231 – 1234. 132. L. M. Lyamshev “Optical generation of sound in liquids”, Proceedings of the 9th All-Union Acoustical Conference, Plenary papers, Moscow, 1977, p. 59. 133. L. M. Lyamshev “On the effect of uneven boundary on optical sound excitation in liquids”, Abstracts of the 8th All-Union Conference on Coherent and Nonlinear Optics, Tbilisi, 1976, v. 2, p. 196. 134. L. M. Lyamshev “Optical generation of sound in a liquid half-space adjoining a solid layer”, Akusticheskii Zhurnal, 1979, v. 25, No. 4, pp. 566 – 574. 135. L. M. Lyamshev “Sound reflection by thin plates and shells in a liquid”, Publishing House of the USSR Academy of Sciences, Moscow, 1955. 136. L. M. Lyamshev “To the theory of optical generation of sound in a moving medium”, Doklady AN SSSR, 1977, v. 234, No. 4, pp. 814 – 817. 137. M. L. Lyamshev “Laser thermooptical generation of sound in a liquid with a free surface”, Ph. D. Thesis, General Physics Institute, USSR Academy of Sciences, Moscow, 1985. 138. M. L. Lyamshev, V. G. Mikhalevich, and G. P. Shipulo “Thermooptical excitation of acoustic fields in a liquid by a periodic sequence of laser pulses”, Akusticheskii Zhurnal, 1980, v. 26, No. 2, pp. 229 – 236. 139. M. L. Lyamshev, V. G. Mikhalevich, and G. P. Shipulo “Thermal excitation of acoustic waves in absorbing media by a periodic sequence of laser pulses”, Akusticheskii Zhurnal, 1979, v. 25, No. 1, pp. 146 – 148.
REFERENCES
351
140. L. M. Lyamshev and K. A. Naugol’nykh “Optical generation of sound. Nonlinear effects (a review).”, Akusticheskii Zhurnal, 1981, v. 27, No. 5, pp. 641 – 668. 141. L. M. Lyamshev and K. A. Naugol’nykh “On sound generation by thermal sources”, Akusticheskii Zhurnal, 1976, v. 22, No. 4, pp. 625 – 627. 142. L. M. Lyamshev and L. V. Sedov “Optical generation of sound in a liquid. Thermal mechanism (a review).”, Akusticheskii Zhurnal, 1981, v. 27, No. 1, p. 529. 143. L. M. Lyamshev and L. V. Sedov “On optical generation of sound in a liquid half-space in the presence of a layer of another liquid at its boundary”, Akusticheskii Zhurnal, 1977, v. 23, No. 5, pp. 788 – 795. 144. L. M. Lyamshev and L. V. Sedov “To the theory of sound generation in a liquid half-space with uneven boundary due to absorption of intensitymodulated laser radiation in it”, Akusticheskii Zhurnal, 1977, v. 23, No. 3, pp. 411 – 419. 145. L. M. Lyamshev and L. V. Sedov “Optical generation of sound in a liquid half-space with an inhomogeneous surface layer”, Akusticheskii Zhurnal, 1978, v. 24, No. 6, pp. 906 – 910. 146. L. M. Lyamshev and L. V. Sedov “To the theory of sound generation due to absorption of intensity-modulated laser radiation in a liquid waveguide”, Akusticheskii Zhurnal, 1977, v. 23, No. 1, pp. 91 – 95. 147. L. M. Lyamshev and L. V. Sedov “On sound generation by a moving optoacoustic source emitting pulses of arbitrary shape”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1979, v. 5, No. 16, pp. 970 – 972. 148. L. M. Lyamshev and L. V. Sedov “To the problem of sound generation by a pulsed moving optoacoustic source”, Akusticheskii Zhurnal, 1979, v. 25, No. 6, pp. 906 – 915. 149. L. M. Lyamshev et al. “On one mechanism of underwater acoustic noise generation in calm ocean”, in: Radiation Acoustics, Ed. by L. M. Lyamshev, Nauka, Moscow, 1987, pp. 46 – 51. 150. L. M. Lyamshev et al. “On one mechanism of underwater acoustic noise generation in calm ocean”, Akusticheskii Zhurnal, 1985, v. 31, No. 5, pp. 709 – 710. 151. L. M. Lyamshev and B. I. Chelnokov “Sound generation in solids by penetrating radiation”, Akusticheskii Zhurnal, 1985, v. 29, No. 3, pp. 372 – 381. 152. L. M. Lyamshev and B. I. Chelnokov “On the influence of particular features of penetrating radiation absorption in condensed media on sound generation”, Zhurnal Tekhnicheskoi Fiziki, 1983, v. 53, No. 11, pp. 2238 – 2241. 153. L. M. Lyamshev and B. I. Chelnokov “To the theory of sound generation due to absorption of intensity-modulated penetrating radiation in a solid waveguide”, Akusticheskii Zhurnal, 1983, v. 29, No. 4, pp. 505 – 514. 154. L. M. Lyamshev and B. I. Chelnokov “Radiation-acoustic microscopy of condensed media”, Akusticheskii Zhurnal, 1984, v. 30, No. 4, pp. 564 – 566. 155. L. M. Lyamshev and B. I. Chelnokov “Basic mechanisms of sound generation by penetrating radiation in condensed media (a review)”, in: Radiation Acoustics, Ed. by L. M. Lyamshev, Nauka, Moscow, 1987, pp. 8 – 26.
352
REFERENCES
156. L. M. Lyamshev and B. I. Chelnokov “Sound generation by penetrating radiation in condensed media”, in: Radiation Acoustics, Ed. by L. M. Lyamshev, Nauka, Moscow, 1987, pp. 58 – 134. 157. L. M. Lyamshev and B. I. Chelnokov “Radiation-acousto-thermal microscopy of biological objects”, Abstracts of the All-Union Conference “Interaction of ultrasound with biological media”, N. N. Andreev Acoustical Institute of the USSR Academy of Sciences, Moscow, 1983, p. 65. 158. L. M. Lyamshev and B. I. Chelnokov “On some peculiarities of sound generation by penetrating radiation in solids”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1982, v. 8, No. 19, pp. 1189 – 1193. 159. L. M. Lyamshev and B. I. Chelnokov “Generation of Rayleigh wave at the free surface of a homogeneous isotropic solid half-space by pulses of penetrating radiation”, Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1982, v. 8, No. 22, pp. 1361 – 1365. 160. L. M. Lyamshev and B. I. Chelnokov “Acoustic detection of high-energy particles in solid targets”, Abstracts of the 3rd All-Union Symposium on Physics of Acousto-Hydrodynamic Phenomena and Optoacoustics, Tashkent, 1982, p. 9. 161. V. A. Malugin and A. B. Manukin “Excitation of acoustic vibrations in solids by low-energy electrons"” Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 1983, v. 9, No. 13, pp. 819 – 821. 162. A. N. Morozov and V. Yu. Raevskii, Foreign Electronic Technology, 1982, issue 2 (284), p. 46. 163. V. S. Murzin “Physics of cosmic rays”, Publishing House of Moscow State University, Moscow, 1970. 164. K. N. Mukhin “Experimental nuclear physics”, v. 1, Atomizdat, Moscow, 1974. 165. N. N. Nasonov “Dynamic effect of beams of fast charged particles in crystals”, Ukrainskii Fizicheskii Zhurnal, 1982, v. 27, No. 12, pp. 1857 – 1859. 166. V. Novacky “Theory of elasticity”, Mir, Moscow, 1975. 167. V. V. Novozhilov “Theory of elasticity”, Sudostroenie, Leningrad, 1958. 168. L. B. Okun’ “Physics of elementary particles”, Nauka, Moscow, 1984. 169. L. B. Okun’ “Leptons and quarks”, Nauka, Moscow, 1981. 170. V. I. Oleshko and N. F. Shtan’ko “On the nature of acoustic waves generated in ion crystals by high-current electron beams”, Zhurnal Tekhnicheskoi Fiziki, 1987, v. 37, No. 9, pp. 1857 – 1858. 171. V. I. Pavlov and A. I. Sukhorukov “Transient radiation of acoustic waves”, Uspekhi Fizicheskikh Nauk, 1985, v. 147, No. 1, pp. 83 – 115. 172. Yu. V. Pogorel’skii “Possibility of excitation of surface sound in semiconductors by modulated absorption of light”, Fizika Tverdogo Tela, 1982, v. 24. No. 8, pp. 2361 – 2364. 173. “Radiation acoustics”, Ed. by L. M. Lyamshev, Nauka, Moscow, 1987. 174. B. Rossi “High-energy particles”, Gostekhizdat, Moscow, 1955. 175. J. F. Ready “Effects of high-power laser radiation”, Academic Press, New York − London, 1971. 176. A. N. Skrinskii “Accelerator and detector prospects of physics of elementary particles”, Uspekhi Fizicheskikh Nauk, 1982, v. 138, No. 1, p. 343.
REFERENCES
353
177. Lord Rayleigh “Theory of sound”, Dover Publications Inc., 2d ed., New York, 1945. 178. “Tables of physical quantities”, Ed. by I. K. Kikoin, Atomizdat, Moscow, 1976. 179. “Technical encyclopaedia. A handbook of physical, chemical, and technological quantities.”, v. 9, ONTI, Moscow, 1932. 180. J. Tyndall “Sound”, ONTI, Moscow, 1922. 181. V. V. Furduev “Reciprocity theorems in mechanical, acoustical, and electromechanical quadripoles”, Gostekhizdat, Moscow and Leningrad, 1948. 182. A. A. Kharkevich “Spectra and analysis”, Gostekhizdat, Moscow, 1957. 183. A. N. Khodinskii, L. S. Korochkin, and S. A. Mikhnov “Properties of ultrasonic vibrations arising in solids under the effect of pulsed-laser radiation”, Zhurnal Prikladnoi Spektroskopii, 1983, v. 38, No. 5, pp. 745 – 748. 184. V. A. Tsarev and V. A. Chechin “Neutrinos for geophysics”, Znanie, Moscow, 1985. 185. V. A. Tsarev and V. A. Chechin, Fizika Elementarnykh Chastits i Atomnogo Yadra, 1986, v. 17, No. 3, p. 389. 186. V. E. Chabanov “Laser ultrasonic testing of materials”, Publishing House of Leningrad State University, Leningrad, 1986. 187. V. M. Chulanovskii “Introduction to molecular spectrum analysis”, Gostekhizdat, Moscow and Leningrad, 1951. 188. “Experimental nuclear physics”, Ed. by E. Segre, v. 1, Inostrannaya Literatura, Moscow, 1955. 189. D. Adlene, L. Pranevivicius, and A. Ragauskas “Acoustic emission induced by ion implantation”, Nucl. Instr. And Methods, 1982, pp. 209 – 210; 357 – 362. 190. U. Ajano “Penetration of protons, alpha-particles, and mesons”, Ann. Rev. Nucl. Sci., 1963, v. 13, p. 166. 191. A. M. Aindow et al. “Laser generated ultrasonic pulses on free metal surfaces”, J. Acoust. Soc. Am., 1981, v. 69, pp. 449 – 455. 192. E. A. Ash, E. Dieulesaint, and H. Rakhouth “Generation of surface acoustic waves by means of a CW laser”, Electron. Lett., 1980, v. 16, No. 12, pp. 470 – 472. 193. G. A. Askarijan et al. “The acoustic detection of high-energy neutrinos”, Proceedings of the International Conference on Neutrino Physics and Neutrino Astrophysics, Baksan Valley, Nauka, Moscow, 1978, pp. 341 – 349. 194. G. A. Askarijan et al. “Acoustic detection of high-energy particle showers in water”, Nucl. Instrum. And Meth., 1979, v. 164, No. 2, pp. 267 – 278. 195. A. G. Bell “Upon the production of sound by radiant energy”, Philos. Mag. and J. Sci., 1881, v. 11, No. 71, pp. 510 – 528. 196. B. L. Beron and R. Hofstadter “Generation of mechanical vibration by penetrating particles”, Phys. Rev. Lett., 1969, v. 23, No. 4, pp. 184 – 186. 197. B. L. Beron et al. “Mechanical oscillations induced by penetrating particles”, IEEE Trans. Nucl. Sci., 1970, v. 17, No. 3, pp. 65 – 66. 198. Y. H. Berthelot and J. Busch-Vishniac “Laser-induced thermoacoustic radiation”, J. Acoust. Soc. Am., 1985, v. 78, No. 6, pp. 2074 – 2082.
354
REFERENCES
199. T. Bowen “Sonic particle detection”, Proceedings of 1976 DUMAND Summer Workshop, Honolulu, FNAL, Batavia, 1977, pp. 523 – 530. 200. T. Bowen “Sonic particle detection”, 15th International Cosmic Ray Conference, Plovdiv, Sofia, 1977, v. 6, pp. 277 – 282. 201. E. Brandis and A. Rosencwaig “Thermal-wave microscopy with electron beams”, Appl. Phys. Lett., 1980, v. 37, No. 1, pp. 98 – 100. 202. S. R. J. Brueck, T. F. Deutsch, and D. E. Oates “Surface photoacoustic wave spectroscopy of thin films”, Appl. Phys. Lett., 1983, v. 43, No. 2, pp. 157 – 159. 203. M. J. Buckingham “The interaction of electrons with lattice vibrations, radiation by fast electrons”, Proc. Phys. Soc., 1953, v. 66, pp. 601 – 604. 204. G. S. Bushman and F. Barnes “Laser-generated thermoelastic shock waves in liquids”, J. Appl. Phys., 1975, v. 46, No. 5, pp. 2074 – 2082. 205. G. Busse “Imaging with optically generated thermal waves”, in: Physical Acoustics, V. XVIII, Ed. by W. Mason and R. Thurston, Academic Press, N.Y., 1988. 206. G. Busse and A. Rosencwaig “Subsurface imaging with photoacoustics”, Appl. Phys. Lett., 1980, v. 36, No. 10, pp. 815 – 816. 207. G. Cargill “Ultrasonic imaging in scanning electron microscopy”, Nature, 1980, v. 286, pp. 691 – 693. 208. G. Cargill “Electron beam acoustic imaging”, in: Physical Acoustics, V. XVIII, Ed. by W. Mason and R. Thurston, Academic Press, N.Y., 1988. 209. D. A. Glaser and D. C. Rahm “Characteristics of bubble chambers”, Phys. Rev., 1955, v. 97, No. 2, pp. 474 – 479. 210. Chia-lun Hu “Spherical model of an acoustical wave generated by rapid laser heating in a liquid”, J. Acoust. Soc. Am., 1969, v. 46, No. 3, pp. 728 – 736. 211. D. G. Davies “Scanning electron microscope”, 1983, pp. 1163 – 1176. 212. D. G. Davies et al., Proceedings of Conference on Microscopy, Society of Photo-optical Engineering, 1983, v. 368, pp. 58 – 68. 6 213. B. A. Dolgoshein “Acoustic signal detection from E = 10 eV particle showers”, Proceedings of 1976 DUMAND Summer Workshop, Honolulu, September 16 – 19, FNAL, Batavia, 1977, pp. 553 – 560. 214. W. B. Gauster and D. H. Habing “Electronic volume effect in silicon”, Phys. Rev. Lett., 1967., v. 18, No. 24, pp. 1058 – 1061. 215. L. S. Gournay “Conversion of electromagnetic to acoustic energy by surface heating”, J. Acoust. Soc. Am., 1966, No. 6, pp. 1322 – 1330. 216. R. A. Graham and R. E. Hutchison “Thermoelastic stress pulses resulting from pulsed electron beams”, Appl. Phys. Lett., 1967, v. 11, No. 2, pp. 69 – 71. 217. M. Greenspan and C. E. Tschiegg “Radiation-induced acoustic cavitation: apparatus and some results”, J. Res. Nat. Bur. Stand. C, 1967, v. 71, No. 4, pp. 299 – 312. 218. H. V. Helmholtz “Theorie der Luftschwingugen in Roehren mit Oeffenen Enden”, Crelle Journ., 1860, v. 57, p. 170. 219. L. Hutcheson, O. Roth, and P. L. Barnes, “Abstracts of the 11th Symposium on Electron, Ion, and Laser-Beam Technology”, Boulder, Colorado, 1971.
REFERENCES
355
220. S. D. Hunter and W. V. Jones “Acoustic signals of non-thermal origin from high-energy protons in water”, J. Acoust. Soc. Am., 1981, v. 69, No. 6, pp. 1557 – 1562. 221. L. Hutcheson, O. Roth, and P. L. Barnes “Laser generated acoustic wave in liquids”, Records of the 11th Symposium on Electron, Ion, and Laser-Beam Technology, Boulder, Colorado, 1971, p. 413 – 420. 222. D. A. Hutchins, R. J. Dewhurst, and S. B. Palmer “Directivity patterns of laser generated ultrasound in aluminum”, J. Acoust. Soc. Am., 1981, v. 70, No. 5, pp. 1362 – 1369. 223. G. Izing “Prinzip einer Methode zur Herstellung von Kanalstrahlen uber Voltzahl”, Akr. Mat. Astron. Ach. Phys., 1925, Bd. 18, s. 17. 224. F. Kakimoto et al. “Acoustic detection of super-gigantic showers”, The 17th International Cosmic Ray Conference, Paris, 1981, v. 11, pp. 426 – 429. 225. G. A. Kenney-Wallace et al. “Nanosecond pulsed radiolysis techniques for the study of liquids using a 600 keV electrons”, Intern. J. Rad. Phys. Chem., 1972, v. 4, pp. 209 – 225. 226. K. Y. Kim and W. Sachse “Observation of X-ray generated ultrasound”, Proceedings of 1983 IEEE Ultrasonic Symposium, 1983, pp. 677 – 680. 227. K. Y. Kim and W. Sachse “X-ray generated ultrasound”, J. Appl. Phys. Lett., 1983, v. 43, pp. 1099 – 1101. 228. G. A. Kenney-Wallace et al. “Nanosecond pulsed radiolysis techniques for the study of liquids using 600 keV electrons”, Intern. J. Rad. Phys. Chem., 1972, v. 4, pp. 209 – 225. 229. E. Lawrence “The evolution of the cyclotron. Nobel lecture in physics.”, Elsevier Publishing Company, Amsterdam, 1965, pp. 430 – 443. 230. J. G. Learned “Project DUMAND and trade-offs between acoustic and optical detection”, AIP Conference Proceedings, No. 52, Long Distance Neutrino Detection, 1978, 1979, pp. 58 – 84. 231. J. G. Learned “Acoustic radiation by charged particles in liquids: an analysis”, Phys. Rev. D, Particles and Fields, 1979, v. 19, No. 11, pp. 3293 – 3307. 232. H. M. Ledbetter and J. C. Moulder “Laser-induced Rayleigh waves in aluminum”, J. Acoust. Soc. Am., 1979, v. 65, No. 3, pp. 840 – 842. 233. C. Lee “Passage of heavy particles through matter”, Ann. Rev. Nucl. Sci., 1963, v. 13, pp. 67 – 102. 234. R. E. Lee and R. M. White “Excitation of surface elastic waves by transient surface heating”, Appl. Phys. Lett., 1968, v. 12, No. 1, pp. 12 – 14. 235. M. Levi et al. “Experimental studies of the acoustic signature of proton beams transversing fluid media”, IEEE Trans. Nucl. Sci., 1978, v. 25, No. 1, pp. 235 – 332. 236. L. M. Lyamshev “Radiation acoustics”, Proceedings of the 1st French Conference on Acoustics, Lyon, 1990, v. 1, p. 7. 237. E. M. MacMillan “The synchrotron proposed high-energy particle acceleration”, Phys. Rev., 1945, v. 68, pp. 143 – 144. 238. P. Marietti, D. Sette, and F. Wanderlingh “Detection of cavities produced in a liquid by ionizing particles”, J. Acoust. Soc. Am., 1969, v. 45, No. 2, pp. 515 – 518.
356
REFERENCES
239. M. A. Markov “On high-energy neutrino physics”, Proceedings of the 10th International Conference on High-Energy Physics, Rochester, 1960, pp. 579 – 585. 240. C. L. Morfey “The sound field of sources in motion”, J. Sound Vibr., 1972, v. 23, No. 3, pp. 291 – 295. 241. T. G. Muir, C. R. Culbertson, and J. R. Clynch “Experiments on thermoacoustic arrays with laser excitation”, J. Acoust. Soc. Am., 1976, v. 59, No. 4, pp. 735 – 743. 242. F. C. Perry “Thermoelastic dosimetry of relativistic electron beams”, Appl. Phys. Lett., 1970, v. 17, No. 9, pp. 408 – 411. 243. L. Reimer “Scanning Electron Microscope”, II, 1979, pp. 111 – 124. 244. R. H. Ritchic “Energy losses by swift charged particles in the bulk and at the surface of condensed matter”, Nucl. Instrum. Meth., 1982, v. 198, pp. 81 – 91. 245. W. S. Roentgen “On tones produced by the intermittent irradiation of a gas”, Philos. Mag. and J. Sci., 1881, v. 11, No. 71, pp. 510 – 528. 246. A. Rosencwaig, Science, 1982, v. 218, pp. 213 – 218. 247. D. Royer and E. Dieulesaint “Analysis of thermal generation of Rayleigh waves”, J. Appl. Phys., 1984, v. 56, No. 9, pp. 2507 – 2511. 248. A. Rujula et al., Phys. Rep., 1983, v. 99, No. 6, p. 341. 249. A. Rujula et al. “Neutrino exploration of the Earth”, Preprint HUTP-83 AO19, 1983. 250. W. Sachse, K. Y. Kim, and W. F. Pierce “X-ray generated ultrasonic signals: characteristics and imaging applications”, IEEE Trans. On Ultrasonics, Ferroelectrics, and Frequency Control, 1986, V. UFFC-33, No. 5, September 1986, pp. 546 – 560. 251. P. G. Satkewicz et al. “Ion-acoustic imaging of surface flows in aluminum”, Rev. Progr. Quantitative Non-Destructive Evaluation, Plenum Press, N.Y., 1986, No. 5A, pp. 455 – 463. 252. J. W. Strutt (Lord Rayleigh) “Some general theorems relating to vibrations”, Proc. London Math. Soc., 1873, v. IV, pp. 357 – 368. 253. L. R. Sulac “The acoustic detection of showers induced by cosmic neutrinos”, Proceedings of AJP Conference on Long-Distance Neutrino Detection, 1979, N. Y., pp. 85 – 98. 254. L. Sulac et al. “Experimental studies of the acoustic signature of proton beams transversing fluid media”, Nucl. Instrum. and Meth., 1979, v. 161, No. 2, pp. 203 – 217. 255. L. R. Sulac et al. “Experimental results on the acoustic detection of particle showers”, Proceedings of International Conference on Neutrino Physics and Neutrino Astrophysics, Baksan Valley, 1977, Nauka, Moscow, 1978, pp. 350 – 368. 256. L. Szillard “Asynchronous and synchronous transformers for particles”, Provisional Specifications L, Pat. Off., 21.03.34., No. 5730, 1934. 257. H. Tateno et al. “Ion beam excited acoustic image and specific element image of teeth, Proceedings of the 6th Symposium on Ultrasonic Electronics, Tokyo, 1985, Japanese J. Appl. Phys., 1986, v. 25, Suppl. 251, pp. 188 – 190.
REFERENCES
357
258. G. Veith and M. Kowatsch “Optical generation of continuous 76-MHz surface acoustic waves on YZ LiNbO3”, Appl. Phys. Lett., 1982, v. 40, No. 1, pp. 30 – 32. 259. P. J. Westervelt and R. S. Larson “Laser-excited broadside array”, J. Acoust. Soc. Am., 1973, v. 54, No. 1, pp. 121 – 122. 260. R. M. White “Generation of elastic wave by transient surface heating”, J. Appl. Phys., 1963, v. 34, No. 12, pp. 3559 – 3567.
ADDENDUM1
Acoustooptics of Penetrating Radiation Studies of optical excitation of sound in gases and condensed media are the substance of optoacoustics, which was developing extensively during the last decades. This is illustrated by the 9th International Conference on Photoacoustics and Photothermal Phenomena [1]. Light beams are beams of particles (photons). In this connection the topic of this book may be considered as optoacoustics (photoacoustics) of penetrating radiation [2]. At the same time, there is another broad field at the interface of acoustics and optics, i.e., acoustooptics. Its topic is investigation of acoustooptic phenomena arising in the process of interaction of light with acoustic oscillations and waves in a substance. Acoustooptics is the basis for development of modern and future devices for microelectronics, signal processing, telecommunications, and informatics. For example, multichannel distributed optical-fiber acoustic receivers are being developed on the basis of optical interferometers and the effects of interaction of light with acoustic waves [3]. Acoustooptic devices using surface acoustic waves are applied widely [4]. Investigation of diffraction, scattering, and interference of waves of penetrating radiation (the de Broglie or matter waves) in the process of their interaction with acoustic oscillations and waves in a substance can be treated as acoustooptics of penetrating radiation, which is also the subject of radiation acoustics. Some fields of acoustooptics of penetrating radiation 1
A considerable time has passed since publication of this book in Russian. New results in the field of radiation acoustics have been published during these years. Some of them concerning acoustooptics of penetrating radiation are given in this section. References to other problems are given as footnotes to the main text. 359
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may be considered traditional now. First of all, this concerns the studies of interaction of X-rays with ultrasound [5]. Other fields are at the initial stage of their development yet. Several years ago the studies were started, which led to the development of acoustic tunneling microscopes [6, 7]. Tunneling microscopy has a very high spatial resolution at the molecular and atomic levels. The first atom interferometer was constructed about five years ago [8]. Probably, it will find its application to acoustic research and measurements at the quantum level. We can give other examples also. Further, we will examine these three sections of acoustooptics of penetrating radiation in some more details.
1. DIFFRACTION OF X-RAYS AND NEUTRONS BY ULTRASOUND IN CRYSTALS The first studies on diffraction of X-rays by acoustic oscillations in solids were performed about 50 years ago [5]. Fundamentally new results were obtained in the 1980s – 1990s (see [9, 10] for example). A common mechanism of acoustooptic interaction based on the modulation of the refraction index of light in the field of sound waves is inefficient in the Xray range. However, in the case of Bragg diffraction the change of the phase differences of the X-ray waves scattered by displaced atoms becomes essential Therefore, the intensity of X-ray reflection turns out to be very sensitive to the level of acoustic disturbances. The regular character of acoustic displacements manifests itself rather weakly at comparatively low frequencies. In the case of high frequency or short waves, an acoustic wave produces in a crystal a macroscopic superlattice with a period equal to the ultrasonic wavelength that leads to strong reflection of X-rays. The amplitude of sound waves can be measured according to the positions of the peaks of reflection intensity. The boundary between low and high frequencies is determined by a relationship between the sound wavelength λs and the extinction length τ of X-rays. The last depends on the energy of X-ray quanta and the material of the sample (crystal). The high-frequency range corresponds to the condition λs ≤ τ and the low-frequency range is determined by the condition λs > τ. The character of the oscillations of intensity of X-ray reflection depends on λs and τ and the sample (crystal) thickness. Probably, the latter prompted Zolotoyabko et al. [10] to study the effect of surface acoustic waves propagating in a thin-film structure at a silicon substrate on diffraction of X-rays. Naturally, they observed a strong change of intensity of X-ray reflection with the change of the amplitude of surface acoustic waves within the high-frequency range. The period and phase of oscillations of the intensity of X-ray reflection are sensitive also to
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weak distortions of the crystal lattice and the parameters of the layered structure. All this opens new opportunities for nondestructive testing of acoustooptic devices for microelectronics. Similar phenomena were observed also in the case of diffraction of thermal neutrons at the spatial lattice formed by acoustic waves in a crystal [11]. As in the case of X-rays in the low frequency range, the increase of the amplitude of acoustic waves (ultrasound) leads to the expansion of the angular interval of the Bragg reflection and the intensity of neutron reflection attains a certain kinematic limit. In the high-frequency range when formation of a superlattice occurs, oscillations of the intensity of neutron reflection are observed. In the case of diffraction of neutrons (in contrast to the case of X-ray diffraction) not only exchange of momentum but also exchange of a very small energy with the ultrasonic wave are essential. Other fundamentally important phenomena are observed too. There is no need here to dwell on these phenomena. They are described by Iolin and Zolotoyabko [11] and in the literature cited by them.
2. SCANNING ACOUSTIC TUNNELING MICROSCOPY Tunneling microscopy and scanning tunneling microscopy are based on the effect of the strong (with respect to the exponential law) dependence of the change of the tunneling electron current between the needle point and the sample surface on the distance between them. Relative scanning of the needle point and the surface provides an opportunity to measure the smallest roughnesses of the surface and perform the topography of the surfaces literally at the molecular or even atomic level. The latter is very important for studies in surface physics. This stimulated a large number of studies on tunneling and scanning tunneling microscopy. Recently, scanning tunneling microscopy was applied to detection of high-frequency (1 – 12.5 MHz) acoustic waves [12]. Acoustic pulses with high-frequency filling applied to the surface of a sample made of an electrically conducting material were detected as the disturbances of the tunnel current. The nonlinearity of the characteristic of tunnel current (the dependence of current on the distance between the needle point and the surface) provided an opportunity to detect the envelope of an acoustic pulse and, therefore, its time and phase parameters and amplitude. It has been determined that the sensitivity of the proposed technique of detection of high-frequency acoustic waves [12] is comparable with the sensitivity of optical methods but its time resolution is much higher. Scanning tunneling
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microscopy proved to be very useful for investigation of the properties of conductor and semiconductor surfaces but less useful in the case of dielectrics. It was revealed that scanning tunneling microscopy using the effects of interaction of tunnel current with an acoustic wave in a sample provides an opportunity to conduct analogous studies of dielectrics with the same high sensitivity. Moreover, it is possible to study and monitor the subsurface structure of materials with the help of scanning acoustic tunneling microscopy. Du Sidan et al. [7] considered various mechanisms of action of acoustic waves on the formation of the “image” of the surface and subsurface structures of various materials in scanning acoustic tunneling microscopy.
3. INTERFEROMETERS USING MATTER WAVES – ATOM INTERFEROMETERS In 1802 Young demonstrated that light could behave like waves at a liquid surface, which propagating and passing through two parallel slots interfere and form a pattern of alternate troughs and crests. Later, in the 19th century Michelson and Fresnel developed optical interferometers on the basis of these ideas and conducted very precise measurements of various physical phenomena using these devices. It was determined that the precision of measurements is limited by the light wavelength. Taking into account the fact that light consists of particles (photons) and the wave nature is inherent also with other particles, physicists just could not avoid thinking about development of interferometers using not photons but other particles and the de Broglie waves corresponding to them. Electrons, protons, and neutrons were used initially as such particles. The last turned to be the most “convenient” also because of the fact that they penetrate into substance relatively well. But the development of an interferometer using atoms, i.e., “heavy” particles, and very short matter waves corresponding to them seemed the most attractive idea. Atoms may be treated as waves only to a certain degree and it is very difficult to reveal their wave nature. Even at relatively small motion velocities, they have a very small wavelength in conformance with the laws of quantum mechanics. Their wavelength is small even in comparison with low-energy electrons and neutrons, for which interferometers were developed earlier [12]. Extremely short matter wavelengths for atoms (matter wavelengths are 1000 times shorter than light wavelengths), and the fact that atoms do not pass through a substance in contrast to neutrons for example, made the development of atom
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interferometers very difficult. Despite big difficulties this development was achieved quite recently, about five years ago. The first two experiments with atom interferometers were conducted in Konstanz (Germany) and Massachusetts Institute of Technology (MIT, USA). The Young experiment with helium atoms was conducted in Konstanz and an interferometer using sodium atoms was developed in MIT. Three diffraction gratings are used in the interferometer with sodium atoms. The first of them splits a collimated beam of sodium atoms into two divergent beams. The second grating is positioned at a large enough distance from the first one (lower at the propagation track of the beams where they have diverged already) and changes the direction of one of the beams in such way that they would converge and interfere. The interference pattern of “troughs” and “peaks” of the type of a standing wave is observed in the region of interference of the beams in the plane perpendicular to the direction of their propagation. In fact, this is the statistical distribution of probability of occurrence of particles in this or that place of the interference pattern. Detection is performed by a running light grating with the period identical to the period of the interference pattern, which moves in the plane of the last. The running grating is created with the help of a He-Ne laser. Material nanostructures were used as diffraction gratings in the experiments discussed above. It was demonstrated in 1986 that it is possible to create diffraction gratings for atomic beams on the basis of standing light waves produced by beams of coherent laser radiation. They can play the role of atomic beam splitters and deflectors like ultrasonic waves serving as optical beam splitters and deflectors. The phenomenon of diffraction of matter waves or particles at light waves was predicted by Kapitsa and Dirac already in the 1930s. Atom interferometers are used for very precise measurements of the effects of rotation, acceleration, gravitation, and other physical phenomena and quantities. Two research teams reported recently successful experiments on development of an atom interferometer analogous to the Sagnac optical interferometer [8]. In this interferometer two atomic beams propagate in the opposite directions along the perimeter of a certain area and meet at the initial place. If the equipment is rotating, the beams have a phase shift when they recombine. The phase shift is directly proportional to the area swept by the beams and the relative rotation around the axis perpendicular to the plane of rotation of the beams (equipment). It was noted for comparison that the Sagnac optical interferometer constructed by Michelson in 1925 for measurement of rotation of the Earth had a perimeter equal to several football fields. Analogous experiments with a neutron interferometer require several square centimeters. The Sagnac effect is the basis of laser ring gyroscopes used widely in aviation. Schmiedmayer et al. [13] reported successful measurements of the so-called refraction index of matter waves
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of sodium atoms in gases (helium, neon, and argon) performed with the help of an interferometer using sodium atoms. Ekstron et al. [14] give the data on measurements of electrical polarizability of sodium atoms with a precision six times higher than that of the measurements conducted earlier using other modern methods. All this can be an impressive illustration of the efficiency of atom interferometers. It is a common opinion that atom interferometers have an excellent future. Atomic sources are rather cheap as well as the devices for splitting and deflection of atomic beams. As for application of atom interferometers in acoustics, there are no reports on this topic yet as far as we know. But there are no doubts that in the near future, atom interferometers may become the basis for development of essentially new acoustic measurement techniques and devices as it happened with optical interferometers or scanning tunneling microscopes.
REFERENCES The 9th International Conference on Photoacoustic and Photothermal Phenomena, June 27 – 30, 1996, Nanjing (China), Conference Digest. 2. L. M. Lyamshev “Radiation acoustics: photoacoustics of penetrating radiation.”, The 9th International Conference on Photoacoustic and Photothermal Phenomena, June 27 – 30, 1996, Nanjing (China), Conference Digest, PL-1.3, pp. 4 – 5. 3. L. M. Lyamshev and Yu. Yu. Smirnov “Distributed optical-fibre acoustic detectors”, Akusticheskii Zhurnal, 1995, v. 41, No. 4, pp. 533 – 546. 4. “Acoustic surface waves”, Ed. by A. Oliner, Springer, Heidelberg, 1978. 5. W. Spencer “Investigation of resonance vibrations and structure violations in single crystals by the technique of X-ray diffraction topography”, in “Physical acoustics. Principles and methods.”, v. 5, Mir, Moscow, 1973. 6. A. Morean and J. H. Ketterson “Detection of ultrasound using a tunneling microscopy”, J. Appl. Phys., 1992, v. 72, No. 3, pp. 861 – 864. 7. Du Sidan et al. “Theoretical study on tunneling acoustic microscopy”, The 9th International Conference on Photoacoustic and Photothermal Phenomena, June 27 – 30, 1996, Nanjing (China), Conference Digest, pp. 542 – 543. 8. B. G. Levi “Atoms are the new waves in interferometers”, Physics Today, 1991, v. 44, pp. 17 – 20. 9. I. R. Entin and I. A. Puchkov “Oscillating dependence of intensity of X-ray reflex excited in a crystal by ultrasound”, Fizika Tverdogo Tela, 1984, v. 26, No. 11, pp. 3320 – 3324. 10. E. Zolotoyabko et al. “Acoustic field study in layered structures by means of X-ray diffraction”, J. Appl. Phys., 1992, v. 71, No. 7, pp. 3134 – 3137. 11. E. M. Iolin and E. V. Zolotoyabko “Interference phenomena in the process of dynamic diffraction of neutrons under ultrasonic excitation”, Zhurnal
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ADDENDUM
365
Teoreticheskoi i Eksperimental’noi Fiziki, 1986, v. 91, No. 6 (12), pp. 2132 – 2139. 12. “Optics and interferometry with atoms”, Ed. by J. Mlynek, V. Babykin, and P. Meyestre, Appl. Phys. B, Special Issue, 1992, v. 54, p. 321. 13. J. Schmiedmayer et al. “Index of refraction of various gases for sodium matter waves”, Phys. Rev. Lett., 1995, v. 74, No. 7, pp. 1043 – 1047. 14. C. R. Ekstron et al. “Measurement of the electric polarizability of sodium with an atom interferometer”, Phys. Rev. A, 1995, v. 51, No. 5, pp. 3883 – 3888.
SUBJECT INDEX
acoustic tunneling microscopy 361 acoustooptics 359, 360 air shower 18, 30, 325 Airy function 100 atom interferometer 360, 362 – 364 Bessel function 57, 63, 67, 209, 220 boson 8, 11, 12 Bragg diffraction 360 Bragg peak 19, 295, 297 bremsstrahlung 16, 32, 257, 299 bubble mechanism of sound generation 4, 30, 31, 35 Buger-Lambert law 48 cascade 18, 19, 21, 27, 30, 227 – 234, 237, 238, 327 – 330 cascade shower 18, 19, 227 Cherenkov direction 202, 205 – 209, 213, 217 – 219, 224, 267, 268 Cherenkov mechanism of sound generation 31 – 33 Cherenkov (acoustic) radiation 2, 31, 32 Cherenkov wave 33 copper sulphate 249, 254 correlation length 58 – 61, 68 cosmic rays 12, 13, 18, 19, 325, 331
cyclotron 287, 288 deformational mechanism of sound generation 277, 278 directivity pattern 51, 59 – 61, 73, 80 – 85, 92, 101 – 103, 116, 150, 151, 155, 156, 160, 201, 210 – 212, 221, 223, 240 – 243, 245 – 247, 252, 254 – 256, 269 – 272, 279, 280 Doppler effect 201, 257, 260 Doppler frequency 208, 209, 213, 219, 224, 259, 260, 269 Doppler wave 260 DUMAND Project 4, 26, 227, 325, 327, 328, 333, 340 dynamic mechanism of sound generation 35 – 37, 148, 152, 157, 232, 234 efficiency of sound generation 4, 5, 23, 31, 37, 39, 72, 160, 161, 163, 164, 191, 192, 212, 213, 227, 238, 252, 255, 271, 275 – 277, 279, 339 electron-acoustic microscope 314 – 318, 323, 325 extensive air shower 18, 30, 325
367
368
SUBJECT INDEX
far wave field (zone) 49, 51, 52, 54, 61, 62, 72, 73, 80, 89, 91, 94, 105, 117, 118, 123, 141, 143, 146, 148, 154, 158, 160, 161, 165, 171, 197, 202, 203, 214, 231 – 234, 240, 243 – 246, 248 – 250, 254 – 256, 258, 260, 262, 266, 280, 288 fermion 8, 10 Fourier integral 64 Fourier transformation 106, 140, 170, 172, 205, 207, 208, 211, 260, 261 Fraunhofer zone (field) 43, 51, 141, 160, 203, 214, 240 GENIUS Project 5, 330, 333, 340 geoacoustics 5, 330, 331, 340 gluon 10 – 12 Grand Unification Theory 8, 11, 12 Grueneisen parameter 25, 103 hadron 8 – 11, 326, 327 Heaviside function 106, 229, 236 inverse piezoelectric effect 38 kaon (k-meson) Kirchhoff approximation 53 Kirchhoff integral 53 latent energy 35 lepton 8 – 12 linear accelerator 1, 287, 297, 298, 302, 332 Love wave 152 Mach number 217 Mach wave 257, 266, 267 matter (de Broglie) waves 359, 362, 363 meson 2, 9, 10, 326, 331, 332 microshock waves 4, 28 monochromatic sound 34, 39, 47, 64, 73, 208, 252, 254
muon (µ-meson) 3, 5, 9, 19, 30, 227, 234, 326, 327, 330, 332, 340 N-wave 130, 135, 273, 287, 291, 335 near wave field (zone) 27, 50, 51, 124, 125, 128 – 130, 197, 198, 230 – 233, 244, 273, 287, 288, 292, 304, 327, 329 neutrino 3 – 5, 7 – 9, 11, 12, 19, 30, 227, 230, 238, 326 – 337, 340 photoacoustic microscope 311 – 315, 318 pion (π-meson) 10, 326, 331, 332 pulsed radiolysis 35 quark 9 – 12 radiation-acoustic microscopy 3 – 5, 309 – 311, 321, 325, 339 radiation-acoustic visualization 309, 321, 323, 325, 339 Rayleigh parameter 55, 56, 60, 61, 68, 69, 87 Rayleigh wave 151 – 156, 162 – 164, 184 – 186, 192, 193, 214 – 218, 224, 227, 237, 271, 273 – 277 reciprocity relationship 42, 43, 45, 142 reciprocity theorem 42, 49, 93 Sagnac interferometer 363 scintillation detector 326 shear stress 232 shear wave 137, 274, 276, 279, 281 signal-to-noise ratio 317, 336 strain tensor 146, 148, 150, 151 stress tensor 160, 161, 165, 172, 174, 176, 178, 180, 181, 193, 198, 214, 229, 230, 232 – 236 strictional mechanism of sound generation 33, 34 surface acoustic wave (SAW) 155, 161, 239, 270 – 279, 359, 360
SUBJECT INDEX
synchrotron radiation 12, 20, 41, 105, 147, 164, 239, 281, 282, 299, 318, 333 theory of underwater explosions 29 thermal (thermoradiation) mechanism of sound generation 3, 4, 19, 20, 24, 26, 30, 31, 33 – 35, 37 – 39, 42, 50, 94, 123, 145, 148, 152, 157, 181, 277, 278, 290, 292, 302, 303, 306, 308, 321, 339
369
thermal relaxation time 41 thermoacoustic array 128 – 130, 150, 155, 156, 172, 181, 245, 246, 249, 254, 287, 303 – 305 thermodynamic generation of sound 24, 47 thermoradiation excitation of sound 5, 39, 42, 50, 55, 72, 75, 131, 239 transition radiation 33, 257 tunneling microscopy 361